Cholas of vijayalaya lineage are called as later cholas to distinguish them from early (sangam) and middle (kochchenganan) cholas.
The nalankilli and nedunkilli titles were applied in sangam age. But it was thought that this usage had vanished in later chola times and the rajakesari - parakesari (notice sanskrit usage) had taken their place.
This does not seem to be true. Aditya I seems to have got the title "Perunarkilli sozhar". Infact there are few inscriptions wherein he is just referred to by this name instead of any other title