What constitutes a legal signature?

Legally, your signature is simply a physical representation of your intent to be bound to an agreement. In other words, you are agreeing to the terms on the document by making your mark. This can be done with a traditional signature, a first or name name only (as you suggest), an X, a thumbprint, or any number of methods. The law now allowing electronic signatures as well, which is helpful considering the amount of commerce conducted online.

The bottom line is that it's your physical representation of assent. It's as simple as that.

A bank will probably not permit you to open an account with just your first name, so no, in that case you would not be able to use just one name. On the other hand, if you did not know how to write, an "X" would suffice with a witness. So, if you are known by two names, your legal signature would require two names.

Your intent when signing, some people put initials or a thumb print. It all comes down to intent if used for fraudulent purposes. You can sign however you like but to avoid arguments/confusion, I wouldn't do so.

Legal signature is that which matches you proof of identity (birth certificate, SS #, Licnese).

No