I suspect what’s going on is that back when many states’ DUI statutes were written, drunk driving was seen as merely a nuisance and was only punishable by a slap-on-the-wrist. So there wasn’t really any reason to separate car from bicycle, horse, etc. But as drunk driving has become recognized as more of an issue, they’ve slowly ratcheted up the penalties without bothering to go back and change the definition to exclude the various low-speed vehicles that realistically are just a nuisance and only deserve the slap-on-the-wrist proscribed by the original statute.
If I wasn’t clear in my post, in my state you can get a dwi while operating anything propelled by a motor on land (boats are a different statute). It can be enforced on public or private property. The penalties do not change.