A log message that is an error is a failure of a system to do something. Typically something can be done to improve matters (a retry, an alert). If you are going to log them look at the logs on a regular basis.
If a client side error was caused from a system outside your direct application it is fine to make another call to store that error,
Give your log messages a useful correlation id. Natural ones are best (the order id being processed) but artificial ones are also useful, if sent through the system.
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