Pronouncing it as a distinct T makes it sound British or pretentious in the US, I agree. There are degrees of shifting it toward the D sound though. It can be too distinctly a d, too, to my ears. In my accent, it’s a sort of tongue tap on the front of the hard palate, as air is still flowing.
Yes, I don’t know the correct way to describe it but it’s a subtle “t”. And there’s always the chance it sounds different to others than it does in my own head 
I used to have a paper ROUT along ROOT 66.
Never has a tiny reindeer set a hoof on my roof. I don’t rhyme hoof and roof, which makes no sense and also (kind of) helps me to understand why others speak the way they do.
Yeah, I was gonna say – that “d” sound in “water” is an alveolar flap (so not your standard dental “d”, but it’s a voiced sound, so it ends up being heard as a “d”, and is standard in at least many dialects of US English. Wiktionary gives two pronunciations for “water” in US English: /ˈwɔtəɹ/ and [ˈwɔɾɚ], the second one of which is the alveolar flap.
This article does a decent job of showing the various pronunciations of “t” in an American accent (it could also be pronounced, for example, as a glottal stop like in “button” for example.)
http://accenteraser.com/blog/the-5-types-of-t-sounds-in-american-english/
I jokingly say it as “theee uh tra”; like an exaggerated, pseudo French thing. Ditto “Brett Faaaavrrrrrra”.
I thought of another one. My sister, niece and I were discussing a recipe. My niece said that both my sister and I pronounce hamburger the same way. We both leave out the first R. Kind of like hambeger or hambuger. I’m not sure how to spell it. And that brings me to something a bit unrelated - we use the word hamburger when other people would probably use the words ground beef. We brown hamburger to use it in a hotdish (casserole!).
I know a few people who call a ground meat sandwich a hamburger, but the meat used to make it is hamburg.
I’m having trouble finding a cite, but I believe that’s the original usage of “hamburger” in English.
It seemed that it had to be prepared since when he was answering questions he lost that accent. He was from Brooklyn originally. I inferred, although I have no evidence, that his point was that how you spoke was unrelated to how you thought. But it was quite striking.
I have been told by a couple of people that I don’t have a Philly accent. I certainly have the characteristics of the Philly accent I described above. My best guess is that I don’t have the South Philly accent. My father had that, but my mother and I grew up in West Philly (and went to the same elementary school and even had some of the same teachers).