Allow me to clarify some issues. Firstly apologies for coming across rude on the coloured issue. I am sometimes abrupt bordering on rude, and it's something I need to work on.
POC/coloured isn't the hill I'm prepared to die on, so I'm not going to spend any more time back and forth. I'll just state my position and leave it at that. Personally I don't use either term, opting instead for either the geographical area or skin colour.
I could be wrong, but it appeared to me that the poster who first brought up the offensiveness of 'coloured', was doing so on behalf of others. Taking offence on behalf of a group (white saviour) is something that raises my hackles, and that probably came through in the response. Again, apologies for that.
For the record, I don't think it's a case of 'asking people what they find offensive', as nobody had ever polled all the black people to know which term they prefer. It was more a case of activists self-declaring themselves to be spokespeople for all others.
As stated, I don't see the difference between POC and coloured people. It's literally the same thing only back to front. However, as it's deemed an offensive term, I don't use it. I wasn't the one who used it to begin with, and I merely commented, albeit somewhat rudely, on the so-called offensiveness of one term vs the other.
Anyway, I'm over and done with that.
As to the question of Jewish-ness or religiosity, I'd compare it to many MNs on the TW issue. While many activists might not like someone saying TW are men, sometimes (always?) truth is far more important than being palatable.
Orthodox Judaism, which from a historic pov is unquestionably the authentic version, believes the Torah is divine. Not 'divinely inspired' but word for word dictated by Hashem.
The five books of Moses (Torah) were dictated word for word by Hashem. All the rest, namely the Prophets and the Writings (Nevi'im & Kesuvim), in addition to the Oral Torah (Talmud), are also divine, but not word for word. However they are all the words of Hashem as related to Moses on Mt Sinai.
If you believe that, there is actually meaning to the term 'religious' Jew. You're following the religion. The Law as dictated by Hashem. However someone who doesn't accept the divinity of the Torah, even if they do say the Sabbath, is merely being a cultural Jew.
For some cultural Judaism is eating matzah balls and gefilte fish, while for others it's also doing a seder and bar mitzvah ceremony. But it's still only about culture and tradition, rather than religion, ie because Hashem commanded it.
Seeing as any other denomination of 'Judaism' except for Orthodox doesn't actually believe in the divinity of the Torah, or in the other 13 tenets of faith for that matter, it is factually correct to state that they're not religious.
For me, someone stating they've converted through reform or any other such denomination and therefore they're Jewish, is as offensive as a man putting on makeup and claiming to be a woman. Judaism is a religion with very specific criteria, and it's not just about how one feels.
Within Orthodox Judaism, there are indeed many streams, and they're all valid. There are many differences based on culture and custom, and even some disagreements on points of law. But they all agree on the basic understanding which I wrote above, that the Torah is divine and all 13 tenets of faith are true.
Hopefully that clears up some matters.