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For the numbers 0-100, only 10 start with 1, exactly as many those start with 2, or 3, etc.

No, 11 numbers start with 1 in that range.

No, 12 numbers begin with 1 in the range [0,100] if that's indeed what the OP meant:

1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 100

I think he most likely meant to exclude 1 and 100, but I don't know.

Kudos for spotting, that. I feel stupid now.