I’m fine with “fur babies” and “Rainbow Bridge” - they don’t bother me at all, although I don’t use those terms myself.
We do refer to ourselves as Mom and Dad to our own dogs though, as in “go give it to Dad” etc.
Everyone is different, but to be honest, none of these words bother me, not even “pet owner”.
Related: Homeowner (HO) is often inaccurate. I have done plenty of sits for people renting.
Although I agree with this in general terms, the implication is the important thing, and in the long run, it’s the “home” part we are referring to. I doubt any sitter has ever thought about whether the people own or rent their home. And we can’t say “Pet Owners” because some sits don’t have pets. Home Owner is the simplest term IMO.
I think homeowner fits. Wherever you live, whether it’s a house, an apartment, a trailer or a yurt, it’s a home. As far as owning or renting,
I don’t think we have to be that specific.
I think ‘home host’ covers all bases
Does it really matter what we are called? 
Nope 
I find it quite remarkable how language has to be changed to suit variable scenarios. It used to be oh so simple. 
So true. If you don’t use the politically correct term, somebody is likely to be offended. It has reached an absurd extreme, IMO.
Yep - we like “host” and “sitter”. We don’t like “owner” for pets either. Since we are all animals, humans included, animal seems wrong too. We tend to use the term that the host uses.
Yep, that would also be my preference. Not only is the term “owner” incorrect (as not all homeowners own homes, and pets by gaining rights can no longer be owned), but in combination with a “sitter” takes me all the way back to feudalism
I also avoid saying ‘pet owner’ as nobody can ever own another living animal (including humans).
We sat for a family who had adopted a rescue cat, and they described themselves as his ‘guardians’. We really liked that, and we use that terminology with our young daughter to explain that we are ‘guardians’ of the animals we care for.
Thinking about that suggestion to use host, I thought I liked it but it keeps making me think of a parasite and its host 
, so NO, I’ll stick with pet owner or home owner.