You may log that or count failures in some metric, but the correct answer is to have a health check on third party service and an alert when that service is down. Logs may help to understand the nature of the incident, but they are not the channel through which you are informed about such problems.

The different issue is when third party broke the contract, so suddenly you get a lot of 4xx or 5xx responses, likely unrecoverable. Then you get ERROR level messages in the log (because it’s unexpected problem) and an alert when there’s a spike.