What's the difference between PCIe linkup and card present?

well, one says "I've detected there's a card", and the second is "I've successfully established a way of talking to it". That's what we can infer directly from the wording!

If you want to understand the software conditions that need to be met for either, Linux is open source, you can read the source that emits these kernel logs

Hardware-wise, in PCIe, detection and link upbring are simply two different things, and the logs reflect that. That's like finding a person (detection) and talking to him (establishing a link) are two different things.

link up means the device is working?

No, it means the link is up. That's necessary, not sufficient.