I only know four religious well. My own, Catholicism, Nichiren Shoshu, and SGI Buddhism. I can spit ball Mahayana Buddhism. Theravada not so much. Muslims I read there is an act and said prayer that one must do and believe in, and they are Muslim.
In my opinion, in all three, when you actually practice the religion and make it your life style. When it shapes your life and the decisions you make, by definition, it is your religion. You know if you are an adherent of that said religion if you not only believe in it but also practice it in heart as well.
What 'counts' (and what doesn't)?
In my faith, if you don't
know your ancestors exist, spirits, and believe in spirits then it's basically psychological belief. Nothing wrong with that just not the same. In Catholicism, you
must take the sacraments in order to be part of the Church. In a Catholic point of view, if you do not take the sacraments, you are not in full union with Christ's Body and Christ himself (the Eucharist). If I were wrong, baptized Christians should take the Eucharist.
In Nichiren Shoshu, you must have the Gohonzon given by their temple directly. You can't get it from any other "denomination" because each denomination thinks the others have fake Gohonzons. Gohonzon is a scroll Nichiren (a Buddhist Ten Tai Monk) wrote on it the summary of the Lotus Sutra which, that, is a summary of all The Buddha's teachings.
With the Gohonzon, according to SGI (lay organization which is a spin off of Shoshu), you can receive benefits from good health to material things without it. You are considered an adherent when you own the Gohonzon and chant to it on a daily basis.
On what basis can someone else say that someone is not an adherent of said religion?
It depends. Some religions may say I am not honoring my ancestors "the right way" so in that sense, although my practice have value, it is not the same or "right" as it would be if I were initiated into their religion.
One every fifty Catholics may tell you bluntly you are not part of the Church if you do not believe in the sacraments of Christ. You can go far from the Church but the Church doesn't run from you, type of thing. They also feel if you purposely break the sacraments, say get married without a prenump, then that, depending on case, could be a cause for the Church divorcing you from the Body. Usually that's in serious cases and, a priest told me, usually for priests and authority of the Church not specifically lay man.
You're not an adherant of Shoshu if you do not have a Gohonzon and attend their services. You're not an SGI member if you don't purchase a Gohonzon. However, the latter does not say you are not a Buddha just not part of the organization hence can't vote and have other privileges that go with it.