Troi's accent is not typically Betazoid based on evidence. Other Betazoids, notably Lwaxana Troi, Devinoni Ral, Tam Elbrun, and Lon Suder, don't.
There is never any explanation given for the origin of her accent in universe.
As you you've pointed out, Deanna's mother, Lwaxana doesn't speak with such and accent. In addition, neither does her father, Ian Andrew Troi, depicted in "Dark Page" around the 25:00 mark.
The bottom line is, there's no in-universe explanation.
Out-of-universe has been covered in the linked question, but with respect to the backstory out-of-universe, one presumes that at the time that Sirtis and the producers decided for her to use the accent she does, the mindset would have been that it was how Betazoids talk. Later, when Lwaxana was added, it was decided that Majel Barrett couldn't or didn't want to play the same accent. From there on, no other Betazoids ever played the accent either. And the last holdout explanation would have been her father. But when he was depicted in "Dark Page" it was decided again (if it was even considered) not to play the accent. This would have been a strange justification anyhow, since he is a human. (Of course this entire paragraph is mere speculation.)