I look at plants more than animals, but generally the rule is if it got where it is without human help (within reason) it is considered 'wild'. For instance, here is how i would classify the following:
a fish caught and photographed where it was caught: wild, even though it is being held at the time
A fish caught and then photographed and tagged at home: captive because it isn't where it would naturally go on its own.
A pet cat roaming a wildlife preserve killing birds: captive, because it is still a pet.
A feral cat roaming a nature preserve or an urban environment: wild. (and yes you may not be able to tell these last two apart! it's tricky).
Cows roaming in a large open rangeland - captive, even though free range
A houseplant: captive/cultivated
A planted tree that is native to where it is planted: captive/cultivated. Who knows if it would have naturally come up in that spot or not.
A planted tree of an invasive species: captive
Seedlings that came up on their own in a lawn - wild. The lawn isn't a natural environment but still the plant got there on its own (admittedly it would be nice to be able to tag them somehow as different from native populations)
A tomato plant that pops up on the garden on its own from some fruit that dropped last year: wild
A spider found in a Vermont house iin winter that would not be alive outside - still wild unless kept as a pet.
Bison reintroduced to the prairie but kept in a fenced pasture - ????????? this one is tricky it came up recently.
A wild hog that may have escaped from a farm but is now totally wild and breeding with other wild hogs - ??????????????
The main thing i consider is range maps and spatial ecology. This is my bias but it's a big part of why we flag things this way. If there are ponderosa pines growing on a mountain naturally, we want to see that they are there. If there are ponderosa pines far from where they are native spreading from a plantation and becoming invasive, we want to see that. Someone planted a ponderosa pine? It's not useless information but we don't want it showing up on the maps of distribution of species (or else the little pin should look different and it should be possible to turn it off). Things can survive way way outside their range when planted and cared for. Our neighbor has a magnolia tree and we are in northern Vermont. It survives because he cares for it. It doesn't make sense for Vermont to show up in the range of magnolia.
Hope this makes sense?