In elementary school, I learned that the round numbers ended with 0. As I progressed, I came to realize that this was equivalent to saying that round numbers are integer-multiples of 10.

Now that you're asking the question, I would generalize that, so that round numbers are multiples of the base.

In binary (converted to decimal), that would be 2, 4, 6, 8, ...

In octal (converted to decimal)l, that would be 8, 16, 24, 32, ...

... and so on.

I also have no problem with negative round numbers.

It strikes me that 0 seems to be a canonical round number in that it's a round number regardless of base.

I wouldn't object if you were to say that round numbers are integer powers of the base (10, 100, 1000, ... for decimal). If your definition doesn't include 0, then I'll expect a good explanation for why not.

But, truth be told, I could learn to live with any definition I can wrap my head around, as long as I can use my elementary school definition in polite company. :)