Yeah, sorry, you're not thinking of it correctly.

Unix time is the number of seconds since Jan 1, 1970. Which, in 2038, will be 2,147,483,648 seconds. This is the same as what a signed 32-bit integer can hold (10^32-1), hence the problem.

Switching to 64-bit can hold this timekeeping scheme for almost 300 billion years.

Note that this is just how Unix decided to keep time when it was being developed. There are obviously many newer implementations that get much more granular than "seconds since 1970" and last longer.. The problem is that many programs have standardized on how Unix does it, so programs know what to expect when calling time().