It is time to tailor the help center pages to be a bit more welcoming and informative, but be careful not to multiply their complexity beyond necessity. The longer and more complicated these become, the more easily they're misunderstood or skipped entirely, then their usefulness drops commensurately.
Always try to resist the urge to be oddly specific in these entries in the absence of real and consistent problems caused by certain types of questions. With possibly two exceptions, Pets SE simply isn't old enough to have experienced these. That said, I'll lend some advice.
It's important to define what this community considers to be a pet - an animal cared for in a domestic setting. That immediately obviates industrial livestock, pigeons in the park, wild birds in the trees behind your home and more.
You can then go on to say that questions about items commonly associated with the care and housing of domestic animals are on topic, provided that they don't fall into the types of questions that aren't a good fit for us. That covers you quite well for now, and the foreseeable future.
I'm not firmly against noting that licensing questions are not on-topic because they broadly depend on your location; I honestly don't see much of a possibility of location-independent questions of that type.
For everything else, please, wait and see if it becomes a real and persistent problem before you make that page any more complicated than it must be. I know that it's difficult to resist the urge to solve possible problems preemptively, but almost everything in our experience has taught us that this is a really bad idea when it comes to new communities.
Let your community scale outward quite a bit more before you really consider that a consensus has been reached. What we've got right now on meta are discussions that happened very early on, by what will be an extremely small representation of the size of this community in six or twelve months.
I'm not advising you not to use the document strategically, but please, make certain that you have a real problem on your hands before you leverage it.