If I were entering the numbers then I guess I should use my own employee number. But I don't bother logging my landing in ACARS as it's only for the company to track my currency.
I do log my landings in my own logbook or more accurately my smartphone app logs my landing for me. So I still do keep track of how many landings I've done.
I'll keep your numbers in mind the next time I fly with a captain who wants to enter my landing for me
Where I work now the FO enters who did the landing after completing the after landing checklist. I assume the FOs done it or not. I'm not particularly interested in micromanaging whether he's done it or not. I guess I'll find out when I get the notice from training to come back early for a PC to reestablish currency.
I guess for the original poster about still logging landings after 2000+ landings - log them if you want - don't if you don't. If you're still looking to get hired somewhere else some airlines are more picky than others but you never know where you are going to end up interviewing so my personal view is to have the logbook prepared where they are going to nitpick (I have no idea which airline nitpicks) because you never know. I bet a lot of pilots from braniff eastern pan am twa never thought they'd ever have to interview again.