Fluit is a very generic term, which can mean any kind of flute or whistle. Dwarsfluit is a specific type of flute. According to wikipedia it's called dwarsfluit (lit. cross flute) because the air flows over the embouchure hole at a 90 degree angle (you don't blow air in the instrument).
A. Yes there is more than one type of flute (western concert flute only being one)
B. Transverse flute is a side blown flute (transverse things going across the shortest axis of something, like a transverse mounted car engine in a front wheel drive car (with a few exceptions like the early Renault 5))
The other types of flute include various sizes i.e.: fife pipe up to hyperbass, end blown flutes (which are mostly non European) and the dreaded pan pipes (technically a cross blown flute)
The use of fluit and flute don't seem to line up across the two languages; a little research I have just done indicates that the translation is correct for how the word flute is generally used in English:
In all cases my reference has been from https://nl.wikipedia.org/wiki/Fluit where I have visited the individual pages, then clicked for an English language version. Essentially, some instruments counted under the broad term fluit are not in general use counted under the term flute.
It might have something to do with avoiding confusion with a whistle; which is also a fluit. I'm not entirely sure, but I think fluit in Dutch entails more subjects than the English flute. Honestly, I don't love this translation though. It's very confusing indeed, I would've said fluit too.
I've played flute for a long time. In US English, we typically just say flute when referring to this flute. But, there are many kinds so, i see why they make the distinction. There are pan flutes and even older styles of flutes. The term they're using refers to a particular style.