What’s the difference between ’ I am the first on the list’ and ’ I am first on the list’ ?
Is the latter option right? Could we use the article before ’ first’ ?

Both are fine and synonymous. In the first, first is a noun and in the second, first is an adjective. Two different ways of saying the same thing.

Thank you for replying . One last thing : why can’t I use the article after ‘in first place, in second place’ etc. ?

I don’t really know, but I suppose that ‘in Xth place’ is idiomatic when speaking of competitions. You can use ‘the’ in other situations: ‘I took the first place I could find in the parking lot.’

It is with some trepidation that I venture into this subject area again, but in another thread we found several similar instances where articles are usually dropped:

“first/second/etc. prize”
“first/second/etc. gear”
“first/second/etc. base”

I suspect there may be others too.

As Mister Micawber says of “place”, I suppose we can label all these as “idiomatic”?