This is unlikely to happen given most cards report monthly with a 30-day delay. Assuming you're able to contest the charges there would be no effect on your rating.
Debit/bank users are definitely the most susceptible.
Additionally as far as I know there is no credit rating penalty for repeated attempted charges above your limit. That sort of information isn't even on any report I've ever seen. The increased balance may impact your credit score, but as you've noted it's unlikely this would make it onto your credit report unless Twilio is rather slow to resolve the issue.
Debit/bank users are definitely the most susceptible.