Seriously? I think the latter is a lot less scary in a lot of ways, and it contains a very specific identifier even if multiple things were happening at once. It also has built-in error correction in case the user reporting the error mis-reads one digit or something that "isn't important". And time zones in the identifier are just begging for misconfiguration problems.

Don't forget that the error phrase is not seen in a total vacuum. It is prepended with the sentence "If you contact us about this error, here is your unique error phrase", which even un-savvy users will recognise as a password (in the children's "what's the password?" sense), that they need to remember exactly and that doesn't itself directly carry information.

On the other hand, if you print out the date/time as part of the error message, the user is just going to glance at it, say, yup that's the current time, and then totally fail to record it (because there's no new information there, right?) or even worse, try to remember it and end up guessing at the time or even the date, making it almost worse than nothing.