Yes I think you are correct.

Nil/null/nothing is different to zero. The former is the lack of anything and the latter is an integer with no magnitude. This is something I'm always having to hammer into our developers' heads over and over again.

The number zero wasn't itself the big thing, it was the introduction of a positional numeral system, which needs zero to represent quantity in a given position, so you can note numbers such as 101 to mean "one hundreds, zero tens, one ones")

Many cultures had at least a word for zero/nothing in the context of numbers, but didn't have a positional numeral system, and in many cases didn't have a numeral notation for it either.

'nihil' translates to nothing, but 'nullus' translates to 'lack of' and was recognized and used as a number, even if the symbols didn't reflect this. Just because zero wasn't typically used when teaching mathematics (i.e. teaching quantity instead of number line) doesn't mean that there was no concept of 'zero' in the numerical sense. Both the romans and the greeks had concepts and proofs based around negativity and zero.

However, modern notation, including negative sign, zero, and algebra, are all rooted in eastern and middle eastern cultures.