From the page that you linked:

The important thing to remember is don’t use possessive apostrophes with any pronouns, either possessive pronouns or possessive adjectives.

If you see an apostrophe with a pronoun, it must be part of a contraction.

its—possessive adjective of it

it’s—contraction for “it is”

The nature of the object doesn’t change which form to use (which should make it easier to determine which is correct), and the correct form is not a debate.

Sure, language changes, but for now that’s the accepted rule.