Definitions can be whatever we want them to be, but I’d argue that the definitions should not be as you posited because eternity is colloquially considered a special type of infinity that has to do with time. Furthermore, something that begins, but doesn’t end, doesn’t make sense to me because something cannot come from nothing, which is what would be required in order to have a something that began, but never ends.
I generally regard infinity as boundless and eternity as timeless (absence of time instead of infinite amounts of it, which is really the same thing). If there were infinite oranges, oranges would be in every location possible to the extent that there would be nothing but oranges because the condition would have to be satisfied that no more oranges could be added to the universe and that would only be possible if oranges were ubiquitous, and if that were the case, then there would be nothing that is not-orange and therefore no contrast or context in which oranges could be said to exist and therefore the oranges would not exist. So the infinite cannot be something that exists.
Hello.
I’m not convinced you have a concept of what existence is.
This dot exists → .
The dot only exists because there is a contrasting background giving it context in which to exist. If there were no background, there could not be a dot and no dot could be said to exist.
So existence is a relation of one thing to another thing.
Abstract existence is not anything we can conceptualize or speak about as real because it would have no contrast or context, unless you could demonstrate how nothingness can be a thing such that we could have something in the context of nothing. But if nothingness were made into a thing, then we’re back to square one because something would exist in terms of the thing that is nothing and, once again, existence is relational instead of abstract.