What are the scientific definitions of "moment (of time)" and "instant"? Are they different with their definitions in everyday language? I also don't know the definitions in everyday language, of course if there are any. I came to this question, while reading a wikipedia page (Instant) which defines instant as "an infinitesimal moment in time, a moment whose passage is instantaneous" and I got confused. Why not say an infinitesimal "interval" in time? Before coming to this, I thought instant and moment are EXACTLY equal in meaning, but I seemed to be wrong.
To answer your question about why they don't call it an interval. An interval is a technical word in some branches of physics, and one that differs from the word interval in mathematics or in everyday speech.
In special relativity the word interval is a shorthand for the spacetime interval which is a number associated with two events (and an event is defined as a where-when: a location in space and also a location in time, a location in 4d spacetime).
So that is a valid reason to avoid using the word interval. That said, the words instant and moment aren't really used differently in any consistent or widespread way. And neither is any different than saying a location in time.
They all correspond to clock readings. They are all trying to refer to a range of possible clock readings smaller than any specific real clock (with a specific real finite nonzero precision) could distinguish.