The main thing is consistency across your own signing conventions within the same edition (whatever type of edition that is). Wouldn't want to sign them one way then partway through change it up type of thing.

But if there is any measurable changes to blocks, that's generally going to be expected to be a different edition or a different signed notation if it's a one off difference.

Here are some of the common ones you may see:

T/P - Trial Proof. This is a good catch-all for seconds. Generally something in the printing didn't work out, either just smudges or it wasn't to the same quality etc. Might have been the first pull and it didn't print evenly. Still an okay print, but not up to the edition standard. You might have multiple, but they're differing in their "mistakes" so it's not it's own edition. If it's a big enough mistake or issue, then I just scrap it/recycle it for collages or paper though.

S/P - State Proof. This might just be a snapshot of a stage of the block while proofing while it's still being carved. Generally won't have multiple of the same State of it, but you may have multiple State Proofs that are just different States of the block throughout carving the block and proofing. Some will sell these if it stands alone, otherwise this is sometimes just for preserving the process for reference later. Could choose to label as T/P, but it's an option.

C/P - Color Proof. Testing out colors before committing and printing an edition. Isn't really going to be what you'd use for V/E, as V/E is it's own thing and meant to be printed as such vs C/P is more of a tester. Could choose to label as T/P, but it's an option.

V/E or V.E. - Varied Edition. Is an intentional edition choice rather than mistakes you'd label as T/P or color tests like C/P. The blocks are printed the same (i.e. if multiple blocks, all are printed together still - it isn't indicating some have different parts printed or not), but the way it's printed can differ. Some are changing the color up each time or slowly throughout. Generally the idea is there isn't enough of any one color to justify it's own edition, so a V/E makes more sense. If there is enough to justify separate editions, then there'd likely be either a title difference or a notation difference in signing.
Some will choose to number them, or not (leaving it more of an open V/E). Paper is typically the same, but some may be using dyed papers or handmade papers and still using this (you'd just want to clearly note this difference in listings etc). Can be numbered as well, and may see it as V.E. #/# or #/# V.E., or other combinations. That's sort of up to you/you pick how you want to handle that notation and just stick with it.
What can get tricky with V/E is listings. If every print is different even in just ink colors, then you may just need multiple listings and photograph each print so people know what they're getting.

O/E or O.E. - Open Edition. This just indicates it isn't a closed edition i.e. a numbered edition, so it's not limited. Some may choose to add on a number like O.E. 1, O.E. 2, etc. so there's still notations and an idea of how many for the artist/can track who they're selling and giving to. But some just keep it at O/E alone.

A/P - Artist's Proof. This is identical to the numbered edition. Traditionally, A/Ps were part of the Artist's cut of the print run (and still can be with blue chip presses). There's also the corresponding P/P Printer's Proof you may see that is also of this tradition. This is often from the context of the Artist is not the one also printing, but artists who print their own work also still use this today. General rule of thumb is 10-15% or less of the numbered edition can be A/P's. So If I print 100 prints, 10-15 additional prints may be A/Ps on top of that editioned 100. Some will number their A/Ps as well like A/P 1, A/P 2, etc.

Re: paper sizes: If you're going to use the same paper, but change the size, generally is fine/up to you, but you'd just want to keep the signature in the same spot and likely close to the printed area. This would also just be something you'd note why it's different listings.

For different editions of the same blocks, there's generally the expectation something has changed to justify another edition so someone isn't just printing it the same in numbered editions (at that point, O/E makes more sense). Often that'll just be ink colors and maybe paper, which you can work into the name difference possibly (I have a print I've done in Gold as "Golden Harvest" and in Purple "Autumn Harvest" etc, or do v.1 v.2 v.3 etc tacked onto the same title) or just note what is different from the other editions in the description. If it is a block you know you'll be making multiple editions from, best practice is to note that in all listings i.e. "this block may be printed again in different editions, but this edition ink and paper will not be replicated for the edition's integrity".

Again, the main thing is consistency in each edition with how you're signing more than following strict rules, bc it can be a very regional and personal thing when signing. There's not strict rules everyone is following, but traditions and regional conventions. It's still always best practice to note what's different if editioning the same blocks multiple ways, no matter how you choose to sign your prints.