So where did Snake get the name 'David' from? Who gave him this name?

The people who raised him.

Major Zero's real name is 'David Oh,' so if Zero really is "The Man with the same codename as Null," he may have named Snake after himself.

But who the hell knows... it sounds more like Snake chose that name at the end to go along with his new life.

Kojima wasn't supposed to continue the series after the end of MGS1 so probably Snake was named after his voice actor name (at least for the PAL version) or it was just random. Everybody has a name...

Incorrect.

Game creation-wise, Kojima named him after the star character of 2001: A Space Odyssey. Otacon's real name, Hal, also comes from the same movie.

Kojima references this at the end of the Otacon ending when Snake says "Hal and Dave, that's a good one. Maybe we should take a trip to Jupitur."

Also, I think Frank Jaeger is talking in-universe here.

In regards to Kojima not wanting to continue the series - that doesn't mean Kojima couldn't add that to the plot if it's not already disclosed, which it isn't.

I agree with Wilson. Snake did say he was raised by many people. This is where he will have got the name David.

I agree. Snake probably wouldn't be using his codename in his private life, especially since he detested everything about his heritage and his former life. Either he got the name David before the events of Shadow Moses (after leaving Foxhound), or this is the name given to him by some of the various people who raised him.

I'd suppose the same person who gave you your name, his "parent"(s) or legal guardian(s).