It depends on the fabric/you can ask. I brought in one delicate item to be washed (a cosplay) and they said they'd wash it in cold and not dry clean it.

The whole point of dry cleaning is not to use water.

Now, sure many dry cleaners will offer laundry services, but that isn’t dry cleaning.

Doesn’t really address my question

No, my button up shirts say dry clean only and I drop then off at the dry cleaner to be dry cleaned, not laundered.

They do both, for cotton shirts they will just launder and press it, which is why its frequently $1-$2 a shirt. For many other fabrics they run it through the dry cleaning process, which is why its $5-7 a shirt.