I would use 'avowed' in the positive sense, as 'confessed' has a negative connotation that's inappropriate for, say, homosexuals or Marxists. Though these days in such contexts even 'avowed' is also rather dated, not very politically correct, because it suggests a rather in-your-face public declaration and a corresponding distaste on the part of the person describing it as such. Now that it's okay to be out of the closet and we're well past the Cold War and McCarthyism, most homosexuals these days are just homosexuals and Marxists are just Marxists, without having to avow or deny anything.
'Self-confessed' works better for admissions of guilt, whether serious (a self-confessed murderer) or humorous, tongue-in-cheek (a self-confessed LEO addict).
For Christians I think I would use 'professing,' if I had to use something. But again, that doesn't come up very much in a society where being Christian is not a stigma and many mainstream Christians are confirmed as young teenagers as a matter of course. Evangelicals who emphasize a moment of conversion and de-emphasize formal creeds would probably just say 'born-again,' though I realize that's not the same on some other grounds.
By the way, I would also suggest deleting the entries
to avow oneself - sich bekennen
to avow oneself a Christian - sich als Christ bekennen,
since 'to avow' is already in there with several translations, the reflexive form is IMO not at all idiomatic, and, as Norbert says, 'profess' is the normal word in Christian contexts.
If a translation is still wanted needed for 'sich als Christ bekennen,' I might suggest something like 'profess one's faith (as a Christian),' 'make one's profession of (Christian) faith,' etc. But all that is really beyond the scope of this thread, sorry.