But "white supremacy" is itself a sort of religion.

I think you are confusing yourself by having dual definitions for both Jewishness and Religion here

You in one place claim Judaism is a religion in the theological sense, with monotheism as a defining feature, rather than solely an ethnic or cultural identity.

But in another you say ethnic and cultural identity alone forming the basis of an ideology is a religion, in which case you would also have to consider atheist secular Jews with a strong in-group preference to be practicing a separate religion of Judaism. But then you'd also have to say they aren't practicing Judaism at all by referencing your previous definitino

Both of these are reasonable definitions if you are consistent, but lead to seeming contradictions when you apply different definitions in different contexts. This is why defining your terms is considered to be so important

Personally I favor this definition analagous to the idea of inheritance in programming

A religion is a specific type of ideology in the general sense, but an ideology is NOT a type of religion.

I.E. A specific type can be an instance of a general type, but a general type in not an instance of a specific type.

Or in other words, a String is necessarily an instance of an Object in Java, but an Object is not an instance of a String. Ideology would be your 'Object', and Religion your 'String'

Any definition will work as long as you apply it consistently to your chosen definition, this paradigm just seems the most intuitive to me for keeping these different concepts with the same word separated.