Thank you so much for taking the time to look in to this, @Hector Vera ! Unfortunately, (and this is what is so weird – and I think what is new behavior), After Effects not only looks at comps that have already been made when deciding what a new comp (with a similar name)’s new sequence number will be, it also now looks at the sequence numbers of the source files them selves. So, in other words, even if I start over and have not created any comps yet, when I make the very first comp of the project (using the example file names from my post at the top, SourceMovie_001, SourceMovie_002, SourceMovie_003, SourceMovie_004, and SourceMovie_005 – again these are the names of the files themselves, no comps exist yet) the first composition will be named SourceMovie_006 (it doesn’t matter which of the source files I used to create the comp, it just names it using the next number in the sequence because it’s taking the source file sequence numbers into account now).
Edit: I also realize now (if you see my post below) that there was a kind of user error involved on my part.
Thank you, again, for trying to help research this!