I'd say "hosting" means making something available to you in the cloud. Model makers don't offer that.
They sell "inference," which is a compute task with real hardware costs for them. The task runs against their model, which is not hosted for you.
So it's similar to why we don't call Heroku or Vercel "hosting" or "compute" because it's more of a service (though "cloud" and "platform" are pretty vague too).
If you have a hosted compiler service and you compiled a program with it, you wouldn't say you "compiler hosted" the source code, no, you still "compiled the source code". In the same way, with ML, if it's hosted or local when you run input through the model you performed "inference", that is the act that has occurred (we can debate the term "inference", but "model hosting" is a worse term).