Jtdylktuy

I'm writing a song in F Major key.

I have a chord progression that goes Like

4/4

C7 - C#dim - Dm - Dm

( V7 - V#dim - Vi - Vi )

After this four measures, I wanted to change key to Dm for a moment.

So thinking the last measure's Dm (Vi) as a tonic (i) for the next four measure,

I went

A7 - A#dim - Bm7-5 - Bm7-5

which is the same ( V7 - V#dim - Vi - Vi ) in Dm key.

But I start to think, is ( A7 - A#dim - Bm7-5 - Bm7-5 ) really ( V7 - V#dim - Vi - Vi ) in Dm key ?

or am I still in F Major key and the actual chord progression is

V7/VI - III#dim - iv#m7-5 - iv#m7-5

what are the details that establish a key change ?

It is subjective. Your feelings define it.

https://en.wikipedia.org/wiki/Key_(music)

... a tonic or tonic chord, which provides a subjective sense of
arrival and rest ...

If you feel that the home base moved, then you feel that the key changed. Someone else might feel differently.

If your real question is, how to create those feelings, i.e. how to establish a key, I think there are questions and answers about that already. Use a proper cadence or something. If establishing Dm is what you're trying to do, then why do you move the bass to B instead of D, and via an A#dim instead of C#dim? If you want your audience to think about an elephant, why say "excavator". You might still be able to make it sound clearly like Dm if you write a strong melody line that disregards the obfuscated chords and overlays something that's strongly in Dm.

Every modulation is a key change. Usually a modulation will have some close relationship to the previous key, like C>Dm, or C>F, or C>G, involving at least a common note or chord, whereas a key change could be a move a semitone or a tone up.

It's not always necessary to change key signature for a modulation - which may even re-modulate back to the original key - but for a complete key change it's worthwhile changing the key sig.

Quite often a piece will drift in and out of, say, C and Am. Certainly more of a mod. than a key change, but not really worth considering as either, unless one wants to be totally academic. It's just what music sometimes does.

As you change from F (C7 is the dominant) to dm this is a false cadence. To establish this new key Dm I propose a normal cadence in the new key Dm, to make the moduation perfect or unambiguous.

Nothing definitively indicates a key change in the kind of situation you are talking about.

Of course it's true that there are many 'textbook' examples of key changes, for example where the change is prepared and fulfilled in a certain way. But in general, there isn't really any one particular thing - or even group of things - that you can point to that 'defines' a key change, and likewise there isn't really anything that defines the 'momentary' key of a piece in the first place.

The nearest thing you will get to a definition of the overall key of the piece will be a combination of key signature, and an instruction as to the key either in the title of the piece, or elsewhere in notation. If there's no such instruction, then at the very least you will have to make your own judgement as to whether the key is major or the relative minor, and while there are many cases where most people would agree, there are some ambiguous cases too.

And of course, you have the same difficulty when discussing key changes, especially (as in your case) between relative minor and major keys. What one person hears as a key change, another might simply hear as a movement to the vi chord.

But I start to think, is ( A7 - A#dim - Bm7-5 - Bm7-5 ) really ( V7 - V#dim - Vi - Vi ) in Dm key ?
or am I still in F Major key and the actual chord progression is
V7/VI - III#dim - iv#m7-5 - iv#m7-5 ..?

It's up to you, as the person analysing the piece, to decide which perspective is more useful.

F major and D minor are so closely related as to be almost the same key. I'm inclined to describe what you're doing as simply sequential imitation within a single key.

Does it make any difference, other that in what labels you put on the chords?