I did a quick google search and found the following excerpts from the rules:
"46.01: As of the round of 16, a club may register a maximum of three new eligible players for the remaining matches in the current competition. Such registration must be completed by 2 February 2023 (24.00CET) at the latest. This deadline cannot be extended.
46.02: Any or all of the players from the above quota of three may have been fielded for another club in the qualifying phase, play-offs or group stage of the UEFA Champions League, UEFA Europa League or UEFA Europa Conference League."
This means it is indeed allowed to register three new players that were signed during the winter transfer window regardless of whether they played for another club in the respective tournament already. The only rule is that not more than three players can be registered for those competitions. I think I remember that Chelsea ran into problems some years ago when they signed a lot of players and had to tell some of them they won't get registered for the Champions League.
I think it all depends on how you interpret the words “three new eligible players” - obviously if a player has already played in this tournament this season, then he is no longer new to it. I think therein lies the nuance.
I wanted to see the details (suddenly the rules really changed), but in your article the link to the UEFA website leads to a page where it says that the material has been deleted
That is weird because the link works perfectly fine for me. It is the link to the English website of the UEFA and the headline of these rules goes as follows:
"...Since 2018/19, it no longer matters whether those players have already played in the competition for another club, even for another club still in the competition."
I think this should be the answer to our question whether a player is allowed to play in the Champions League for the new club if he has already played for another club in the same competition. The deadline to nominate the newly transferred player is as mentioned above (February 2nd in 2023 for 2023). I didn't check what the deadline is for 2024, but we basically now know that three players can be nominated regardless of their potential appearance in the competition for another club.