Never in 30 years professional flying have I ever heard of anyone recording t/o - landings, nor of any requirement to do so or even seen a log book in which you could do so (though there are evidently some out there). How do landings have anything to do with P1u/s? A conceptual problem there I think. I suppose in long haul it might be something to keep an eye on but for most pilots its a non issue. Anyway, recency in professional aviation is the company's job, surely?
In a PPL or single pilot operation every landing is yours.
Never heard of a requirement for students to do this either - an appropriate remark is usually put in the remarks column describing the ground covered in the sortie but listing numbers/types of landings is a bit ott.