I was just wondering why in Arena or Daggerfall, if I am not mistaken, was there no Imperial race to choose from? I was thinking if it had to do with something about the Lore or was it that Bethesda simply just came up with it for a race in the later installations of TES?

I'm not sure, but at least in Arena, the game won't let you travel to places in Cyrodiil. It won't even show anything on the map except the major roads and the Imperial City itself.

So they either had not made up their minds back then what the heart of Tamriel should be like...

... or they started off with the idea of a giant salad bowl where you can find residents of all the surrounding places (Argonians, humans, elves, ...) and there wasn't anything special about it.

I heard that Morrowind introduced the race of Imperials, but weren't they already in Battlespire or even Redguard?

I think they started out in Redguard because of Cyrus dealing with an Imperial governor or what ever. They became playable in Morrowind however.