Purchasing a firearm as a gift, however, is not a "straw purchase." If someone, say your parents, bought a handgun as a gift for you and you were legally able to own it (as here in Florida), I doubt most FFLs would bat an eye. Heck, you can even get a CCW at 18 here. Buying a handgun for your kid as an 18th birthday present is pretty much a tradition in some parts of this state.
Generally, "straw purchases" are whenever someone buys a gun on behalf of someone else who is legally prohibited from doing so.
A 20 year old (at least in Florida) is legally prohibited from buying from an FFL, but NOT from buying from a private seller. I suspect most FFLs would avoid this type of situation if they knew about it, but if they didn't, I don't see where any law would be broken. Once the older party owns the gun, what he legally does with it is his business and the FFL is no longer involved.
As a practical matter, if it's legal for you to own the handgun and you never do anything illegal with it, why would anyone ever have cause to investigate how you acquired it?