Does the term "culmination" refer to crossing the local meridian, or reaching the highest point in the sky?

The Wikipedia article on "culmination" defines the term as

the passage of a celestial object ... across the observer's local meridian.

It later says

Except at the geographic poles, any celestial object passing through the meridian has an upper culmination, when it reaches its highest point (the moment when it is nearest to the zenith)... . The time of culmination (when the object culminates) is often used to mean upper culmination.

But as discussed here, the Earth's axial tilt causes the Sun to cross the local meridian at a different time from when it appears highest in the sky (closest to the zenith).

Is it the case that

1. "Culmination" and "upper culmination" actually mean "closest to the zenith"? or

2. "Culmination" and "upper culmination" actually mean "meridian crossing"? or

3. "Culmination" means "meridian crossing", "upper culmination" means closest to the zenith, and an upper culmination is not a culmination? or

4. For the Earth, the meridian crossing and the closest-to-the-zenith occur close enough together than there is no standardardized definition of "(upper) culmination"?

I would find case #4 to be a bit suprising, since the difference between the zenith and meridian crossing can be nontrivial - more than 20 seconds at the equinoxes.