By the time you're having this conversation with a BA, the DB is not going to be changed because they made some offhand reference to a different DB. They said something technically wrong, but in a way that doesn't have any meaningful effect on what they were trying to say: as such, to nitpick a detail like that may give the impression of trying to reinforce your superiority by pointing out tiny technical mistakes in their speech. So just let it go :)
Giving an exact specification is a statement that you want it this way. It is a property of an executive position to give correct instructions to the subordinates. If they are not capable to do that they are simply not qualified for their job. On the other hand: If you want feedback for your specification, ask for it. Giving instructions means "you implement it this way and I am personally responsible with my honor that this is indeed what is needed".
One minute of googling can be expected from BAs, too.
> They're phrasing the requirement in terms of their best understanding of the implementation.
Then they should say "to my knowledge MySQL is a popular database that I know of and is to my knowledge used a lot in the company, but if you know of a better implementation (I don't know much about that - I'm no computer scientist) use it instead.".