Sign language only has a few (thousands) of signs. For the rest the signer finger spells the word they need, if they will use that word again they make up a new sign (often the first letter in a specific position). If a word is commonly used it will eventually enter the language in that city, and may or may not spread to other parts of the country/world. Obviously the finger spelling is the word in the spoken language not sign language.
Warning, don't make up words in sign language until you are somewhat an expert. I know of a novice who needed to talk about a person named Ryan, so they gave Ryan a sign - not realizing that sign was already used for retard.
Plausible deniability here.
I'm not sure where this information comes from, but doesn't quite seem correct. A sign is much more involved than simple vocabulary, and qualified signs exist outside the concept of a one-to-one, sign-to-word relationship.
There are five parameters to any sign: handshape, location, movement, palm orientation, and non-manual communication (e.g. grammar, tone, and emotion). These parameters are all in use for every signed vocab word you can think of, but they're also used in idioms, whole phrases, classifiers, depictions, and even lexicalized English words that have incorporated fingerspelling elements with other parameters to create complete, separate, non-vocabulary signs of their own (e.g. KILLED, PIZZA, BUZZ, COMMON).
And there can be disagreements in the Deaf community about, for lack of a better word, 'purity' of the language. Is it 'okay' to just sign English words with English grammar? Is there an ethical or moral imperative to maintain ASL as a language and not just a translation of another language? And then you have one person telling another person that there's something wrong with how they communicate their native language! Language overlapping with culture overlapping with disability (or different-ability?) overlapping with people being people.
There's no easy answers. But it's fascinating.