I'd suggest there's credence for all of these

• Your fuel pump overheating if it runs dry

Unbeknownst to most, the fuel in the tank cools the fuel pump in most vehicles. The pump sits in a bath of fuel for just this reason. Seems counter intuitive you'd stick an electrical device into something as flammable as gasoline, but it works just fine because there's not enough oxygen in the tank to allow ignition. If the tank runs dry or near dry, heat can build up in the fuel pump which will shorten its life.

• Water condensation in the empty areas of the tank, freezing or causing corrosion

Of the four, this is probably the least likely (at least the freezing part), though I could see it happening. In any area of the world which uses ethanol (like most of the US), the fuel blend tends to collect moisture through absorption. It does this from the air which is vented into the tank from the outside. The more air you have in your tank, the larger possibility there is something like this might happen. By keeping your tank more full, there is less space in the tank for the fuel to absorb water from, which means there's less of a chance of freezing.

Corrosion is still a factor and can be caused from excess moisture. It takes time, but it does happen. Again, keeping your tank closer to full helps preclude this issue.

EDIT NOTE: Vehicles since around the 90's have mostly been fitted with non-metal fuel tanks, so will not suffer from this like they used to. (Thanks to @FreeMan for bringing this up in the comments.)

• Sediment accumulating and blocking your fuel filter

As per the previous portion, if you get excess moisture (which causes corrosion) the bits/pieces which flake off due to this will help to clog your filter over time.

EDIT NOTE: Along with the previous question's edit, sediment may not be prevalent from rust flaking off like it used to be with metal tanks, but sediment still does occur in the tank. Running a tank low may allow whatever sediment to exist to restrict or possibly even clog the pre-pump filter.

• Making sure you'll never run out of fuel, leaving you in dire straits

Having grown up in the mountains of Montana where snow is a factor for four to five months out of the year, this is an extremely important thing to remember to do, for a couple of reasons.

First of all, if you should lose control of your vehicle and run off the road or get stuck for some reason, it may be a while before someone can come and rescue you. If you run out of fuel while you're waiting, you could literally freeze to death. We were taught not to run your vehicle any more than 10 minutes every hour, which should give you plenty of time if you have fuel in your tank.

Secondly, Having more fuel in your tank provides more weight to your vehicle, which helps with traction. This little extra traction can help you stay out of trouble in the first place.

Your fuel pump overheating if it runs dry

This is indeed a concern, as with all other fluids in the car. Most car fluids serve dual purposes-- cooling as well as lubrication. You're not supposed to run low on anything. More volume provides more dissipation of heat.

Seals also start to dry out and crack when dehydrated. This is a concern when storing a car for a long time, which is one of the reasons why you're supposed to completely fill it with gas (and stabilizer) before storing it. This is also why the air conditioner in cars formerly owned by geriatrics tend to be broken (they never use it, so everything dries out, the pump degrades and the refrigerant escapes).

Sediment accumulating and blocking your fuel filter

I received similar advice from my dad ("never let the tank run to empty"); this is probably leftover advice from the 70s-80s when gas tanks were still made of metal and subject to corrosion. Rust flakes could clog a filter for sure.

Fuel intake is on the bottom of the tank. Like sand in your bathtub, any sediment introduced is going to naturally settle around the drain and get sucked into the filter as fuel is drawn through it regardless of volume. That's what the filter is for-- it's a consumable item. You're supposed to replace it according to your maintenance schedule anyway.

Myself, I don't live in a dusty climate so I just clean the gas cap and fuel nozzle when refueling to prevent dirt getting in in the first place.

Running out of fuel unexpectedly, leaving you in dire straits

Also a concern if you live outside a serviceable range of roadside assistance.

On the flip side, the less fuel you're carrying, the better your gas mileage (at the expense of traction).

Half full seems way over the top. I normally refuel somewhere between 1/4 and shortly after the low-fuel warning coming on. If I were out in the back of beyond I would always make sure I had plenty of fuel to reach the next filling station.

I have heard (also had disputed) that if a diesel car were ever to completely run dry the resulting lack of lubrication would completely destroy the high-pressure CRDi fuel injection pump. So diesel cars are programmed to shut down and pretend to have run out of fuel while there remains a litre or two of diesel in the tank (but with the gauge below empty). Given what running even briefly on gasoline does to an injector pump, I can certainly believe this.

A very low fuel level may combine with vehicle motion to swirl any sediment out of the tank and into the fuel line, so a filter blockage is also a plausible concern.

But both these concern running on the dregs, not running below half full.

BTW there is a small but non-zero economy penalty for filling up early. The average weight of fuel you are lugging around will be higher, so your fuel economy will be worsened.

I used to own a car that ran on LPG (Honda Accord). This meant my LPG tank was my main fuel source, and the full-size petrol tank was used only when starting the engine, and as a backup. I tended to put about 20 litres of petrol in every 3 months, so the tank was never more than 2/5 full. When I sold the car it was 12 years old and ran without issue, suggesting points 2 and 3 are not major factors.
I live in .nl where the climate is temperate and fuel is of high quality, which would influence those points.

With all the good input, and with a bit more research, I'd like to answer my own question. So, here were the concerns I raised on letting a car's fuel tank get close to empty:

Concern 1: your fuel pump would overheat if it ran dry

In-tank fuel pumps are cooled by the flow of gas around the motor:

Source: www.aa1car.com

So, there is a valid concern that running the pump dry will leave the motor uncooled, causing damage. However, pumps seem to draw less than 10 amps (see here, here, and here) which at 12 volts is probably less than 100 watts. That isn't much power; it would take at least fifteen minutes for it to heat a cup of water to boiling. You probably shouldn't continually try to crank a car with an empty gas tank, but I doubt a second or two of fuel-free running would overheat a pump.

Concern 2: water would condense in the empty areas of the tank, freezing or causing corrosion

Water can get into your fuel tank in two ways: mixed into the pumped gas when filling, or as humid air being pulled in as fuel is being pulled out and used. For the former, the amount of water is just how much gas you pump; it doesn't matter how you split it up into fillings. For the latter, as cbeleites noted, emptying a tank halfway twice pulls in exactly as much air as emptying a tank completely once, so again you'll get the same amount of water pulled in no matter how you split that up into fillings.

As freeman noted, OEM fuel tanks these days are plastic, not metal, so corrosion isn't going to be an issue. In terms of freezing, if there's ethanol in the gas then the water will likely blend with the gas, so it won't freeze. Any water that doesn't mix in will sink to the bottom and probably get pulled through the fuel pump. This could be a problem, but the magnitude of the problem won't depend on whether or not there's lots of gas floating on top of the water.

Concern 3: sediment would accumulate and block your fuel filter

This doesn't seem likely to me, assuming you buy from semi-reputable gas stations. But, if a gas station does sell you dirty gas, or if someone spoons mud into your gas inlet, it would sink to the bottom and likely be sucked into the pump whether or not there's lots of gas in the tank. Yes, the sloshing of a less-than-full tank would move the sediment around and make it more likely to get pulled into the pump, but unless you drive a limo on really smooth roads, or keep your tank almost completely full, that's still going to happen. And, as ivan noted, that's what fuel filters for; they are designed to capture silt and be replaced at regular intervals.

Concern 4: you might run out of fuel unexpectedly, leaving you in dire straits

This is a real concern, but it has to be evaluated based on where you live and what your priorities are. I live in the New England suburbs, where there's a gas station every few miles, and even on the highway I'm clear about how far I need to go and how far my tank will take me. And, if I make a mistake and do run out of gas, it will be inconvenient, not dire. If you want to make sure you can always drive a couple of hundred miles at a moment's notice, then you should keep some gas in your tank. If not, don't worry about it.

Postscript: the benefit of filling only when your tank is empty

My car gets about 25MPG, and has a 14-gallon tank. If I drive it 120k miles then that's 4800 gallons of gas over its lifetime. If I fill my tank up completely each time, that's 340 fills; if I fill it halfway then that's 680 fills.

If it takes me five minutes to get to and from the gas station (ignoring the time spent actually standing at the pump), then 340 fills would take about 28 hours, and 640 fills would take about 56 hours. So, by only refilling when empty I'd save about 28 hours over the life of my car. Seems worthwhile to me.

Thanks for everyone's input! Additional feedback welcome!