In your example, the token contains the timestamp.
With prefixed-api-key, the hash of each token is stored in the database, so the timestamp can easily be added there.
Most of the time I don't see the utility in the client having the timestamp, outside of a scenario where you have a third party validate on their own (e. g. JWT w/ RSA keys). The best way you see if a token has expired is by trying to use it.