This might depend on your definition of “playboy” (see also: Is Cater attracted to both genders?)
If your definition of “playboy” is “someone who dates a lot of different people,” then “playboy” does not apply to Cater, as far as we know.
What we do know is this:
His family moved around a lot when he was a child so, at some point, he started intentionally keeping his interpersonal relationships shallow rather than try to forge meaningful connections:
“That’s why I always tried to be on good terms with everyone, rather than forge strong bonds with a chosen few. Like a circus performer who has a grand old time with people from around the world, and then packs up and moves on.”
While never stated outright, it is a common theory that this is why Cater keeps Trey at arm’s length during Wish Upon a Star:
Trey is already his closest friend, so Cater might be keeping up a buffer zone of secrets like what his true wish for the Star Sending is so that they do not become any closer, and their inevitable parting will be less painful.
Lilia tries to encourage Cater, saying that, in deciding not to get too close to any one person he may be the wisest one there, but Cater is not convinced that Lilia can truly empathize:
“Lilia’s developed cherished relationships while living in the same place his whole life. There’s no way he could ever understand how helpless I feel.”
Another thing we know about Cater is that he might be feeling the loss of one, specific friend in particular! He tells Silver that he remembers making a friend at a new school one time, only to immediately learn that he would be moving away again the next month.
It is interesting that he mentions this one friend in particular–were they the catalyst? Was that the point when he decided that he would never get close to another person? We do not know.
If you are seeing “playboy Cater” coming from the JP side of the fandom, it is possibly because his speech patterns are that of a チャラい (charai) character on JP!
Localized as “shallow” on EN, it is the reason why Eliza rejects Cater during the Phantom Bride event.
They are characters that talk/look flashy, seem shallow, and tend to flirt with everyone equally rather than picking someone specific to pursue.
This survey of 3,324 people voted the top three チャラい anime characters as Moroboshi Ataru (Urusei Yatsura), Saeba Ryo (City Hunter) and Lupin III, to give you an idea of the チャラい character archetype.
And a common theme of Twst is: appearances can be deceiving!
Vil and Lilia seem to suspect Cater of having “a morose side that the rest of us never see” hidden by his sunny disposition, but Cater claims otherwise.
Much like Epel appears delicate but is actually the opposite and Lilia appears young but is actually old, it is possible that Cater intentionally acts shallow while actually he has a complex inner world–he is flighty not because he is trying to attract people, but because he is trying to keep them away.
Twisted Wonderland Reference Note 