Not really any more than you already have. In Indiana, they already tax folks higher for homes you don't personally live in (no discounts here). Not only that, but rental income is income: This is a business you are running, after all, and you already have to file taxes on that income (and expenses). It truly wouldn't be that difficult to register a house as a rental and giving it the discount if you think it should apply.

I'd personally only give it out to folks with longer terms: The place should still be someone's primary residence, and only give a discount if it has been rented out x months out of the year. This would be to discourage airBNB units (for example) and encourage landlords to make their units affordable and attractive to the residents of the area.

Would that not by definition be a place the owner doesn't live in?

Yes. And I would suggest that we shouldn't tax renters more than owner-occupiers. Renters are already poorer on average.

Yes, and some states already charge more property tax for rentals because the owner doesn't reside primarily at the address.