[brenns10]

You missed the initial part, where the signer finger spells the concept the first time they use it, and then they come up with a sign representing it. By doing so, they've shown the other person what sign they're using and its meaning.

It's like if you write a long jargon phrase and then define an abbreviation in parentheses. Next time you can use the abbreviation. If the abbreviation becomes well known enough (especially in certain groups) then you may be able to omit the definition altogether, especially if you already know your audience.

I'm not sure about schools "manufacturing their own gestures" but sign languages tend to have regional dialects and shared jargon, much spoken dialects and jargon. It could be that these signs are simply regional variations, or that a single sign hasn't become dominant.