For instance, why are moles and decibels considered dimensionless, but kg and meters aren't? Or, in other words, what exactly is a "dimension" in this context? Is just about the system of units? Like, if we use a unit system where c=1, does that make speed a dimensionless quantity in that unit system? Or is it something about the quantities themselves?

I did see the answers to this question but none of them really pin down a specific definition for "dimension" in the context of dimensional analysis or specific criteria for deciding if a quantity is dimensionless or not, which is what I'm looking for.

I also found this question, but the two answers aren't consistent with each other or very helpful -- one says that moles aren't actually dimensionless and then just talks about how we should be more explicit when writing units, and the other sort of vaguely talks about not all base units really being fundamental (which is true, in a sense, but doesn't really answer the question).

For instance, why are moles and decibels considered dimensionless, but kg and meters aren't? Or, in other words, what exactly is a "dimension" in this context? Is just about the system of units? Like, if we use a unit system where $c=1$, does that make speed a dimensionless quantity in that unit system? Or is it something about the quantities themselves?

I did see the answers to this question but none of them really pin down a specific definition for "dimension" in the context of dimensional analysis or specific criteria for deciding if a quantity is dimensionless or not, which is what I'm looking for.

I also found this question, but the two answers aren't consistent with each other or very helpful -- one says that moles aren't actually dimensionless and then just talks about how we should be more explicit when writing units, and the other sort of vaguely talks about not all base units really being fundamental (which is true, in a sense, but doesn't really answer the question).