Every egg the game talks about is held by a guardian. The current official step is "Steal eggs from guardians." That is a guarding role, not a bestiary.
Escape is in one wording only
The longer recorded description says: train your backflips, jump higher, steal rare eggs, and escape from their powerful animal guardians. "Their" means each egg has a guardian, and the escape is written as part of the steal.
The current official description does not describe a chase. The two lines do not contradict, but only one of them mentions getting away. The first time you take an egg, watch what that animal does.
No species roster
The description does not name guardians, count them, or assign them to biomes or egg tiers. In one recorded copy, the phrase "powerful animal guardians" sits beside dragon, wolf, and lion emojis. Those emojis are a hint in that copy, not a confirmed species list. Do not plan a route that assumes a wolf owns one biome.
The same longer text lists Common through Mythic eggs and does not say the guardian or the escape gets harder at higher tiers.
What to do with that
Treat every egg as owned. Take it, then be ready to leave if the animal reacts. There is no published catch penalty, speed, or tool that stuns a guardian. After you are clear, the longer text says the egg still has to reach your base before it is yours to hatch.
The egg tiers and the return trip are covered in Steal an egg.