Short answer: they're both fine, and you clearly already understand when to use which.

Use of with a noun phrase, to with a predicate, and that for a clause. You gave very good examples of all three in your question. The original reason the to usage was frowned upon is that convince was supposedly only for thoughts and ideas; if something someone said resulted in you actually doing something, then you were persuaded rather than convinced. This rule is rather old. Here's what Merriam-Webster's Dictionary of English Usage has to say:

To sum up: long ago persuade became established in a use connoting mental acceptance without following action -- a sense Richard Grant White thought should be reserved for convince. Persuade still has this use, often with the same of and that constructions regularly found with convince. Sometime around the middle of this century, convince began to be used to connote mental acceptance followed by action, usually in a construction in which an infinitive phrase follows the verb [that is, with to]. This construction is now a fully established idiom. The earlier usage writers who tried to fence off persuade from convince and the later ones who tried to fence off convince from persuade have failed alike. And in another generation perhaps no one will care.

You can read the rest of the entry on page 297 on Google Books. They also give attested usages from the New York Times, etc.

Eduardo may have a point about the British/American thing. Just a quick search of the BYU-BNC British National Corpus and the Corpus of Contemporary American English, which contain millions of words from published material gave this for the BNC:

• 7 hits for convince him to
• 12 hits for convince him of
• 39 hits for convince him that

and this for the COCA:

• 140 hits for convince him to
• 29 hits for convince him of
• 131 hits for convince him that

Keep in mind that each of these usages differs also in meaning, so be careful what conclusions you draw from these numbers.

You may also be interested in the Google Ngram Viewer.

Short answer: they're both fine, and you clearly already understand when to use which.

Use of with a noun phrase, to with a predicate, and that for a clause. You gave very good examples of all three in your question. The original reason the to usage was frowned upon is that convince was supposedly only for thoughts and ideas; if something someone said resulted in you actually doing something, then you were persuaded rather than convinced. This rule is rather old. Here's what Merriam-Webster's Dictionary of English Usage has to say:

To sum up: long ago persuade became established in a use connoting mental acceptance without following action -- a sense Richard Grant White thought should be reserved for convince. Persuade still has this use, often with the same of and that constructions regularly found with convince. Sometime around the middle of this century, convince began to be used to connote mental acceptance followed by action, usually in a construction in which an infinitive phrase follows the verb [that is, with to]. This construction is now a fully established idiom. The earlier usage writers who tried to fence off persuade from convince and the later ones who tried to fence off convince from persuade have failed alike. And in another generation perhaps no one will care.

You can read the rest of the entry on page 297 on Google Books. They also give attested usages from the New York Times, etc.

Eduardo may have a point about the British/American thing. Just a quick search of the BYU-BNC British National Corpus and the Corpus of Contemporary American English, which contain millions of words from published material gave this for the BNC:

• 7 hits for convince him to
• 12 hits for convince him of
• 39 hits for convince him that

and this for the COCA:

• 140 hits for convince him to
• 29 hits for convince him of
• 131 hits for convince him that

Keep in mind that each of these usages differs also in meaning, so be careful what conclusions you draw from these numbers.

You may also be interested in the Google Ngram Viewer.