IMO "seduce" has the negative connotation that you are somehow manipulating a "target", whereas "charisma" is more positive; it means your behavior attracts positive attention -- that people find you compelling and interesting, that they will want to follow your lead. (Granted, in concrete terms these may not always be so different -- but to say that is to accept a cynical frame.)

I understand "rizz" as entirely a short form of "charisma" with all the same connotations -- in which case I wouldn't focus it to "romantic interests" or to the goal of "ultimately seduce[ing] them" -- but it's possible that younger people use "rizz" differently than I use "charisma".

The word "charisma" is a little gendered. You are more likely to hear a man described as "charismatic" than a woman. A woman is more likely to be -- well, honestly, the word would probably just be "hot" (or "sexy" -- as used here for Garibaldi), which would place a lot more emphasis on appearance; though to focus more on behavioral aspects, one might hear (I'm reaching here) "effervescent" or "fascinating". I suppose "magnetic" is also an option, though that seems pretty gender-neutral to me.

The slang "game" you mention seems intermediate in connotation to me. The meaning is closer to "ability to seduce", but the connotation is not quite so negative. The implication is that "the game" is simply part of life, neither good nor bad; it is what it is.

Of course, the connotations of "rizz" or "charisma" are less sexual overall. Possibly some amount of what I'm calling "positive connotations" or "negative connotations" simply reflects that, and our society's Judeo-Christian negativity towards sex (which in "conservative" places forces LGBT people at least a little into the closet; and in "liberal" places survives primarily for straight men, the bastards).

Put another way, "charisma" or "rizz" is work-safe, indeed desirable in a CEO (and will likely be rewarded in most employees); whereas "seduction' is not work-safe.