This is a big topic in aviation. But the example you link to is about ACARS messages which are sent to the ground via data link, not visible to the crew.

The ECAM system is what displays issues to the crew. There is a lot of logic there on what to show and what to inhibit. For example some errors aren't shown during takeoff since it's more important to focus on safely starting the climb, they would only be shown once past 400ft altitude.

There is also a priority order based on criticality, they don't just show up in chronological order. Engine fire for example is a red warning and would always come above a yellow caution message. It's designed in a way where the results of the major problem (e.g. engine on fire, and then loosing electric power from that side) are lower priority than the problem itself.