Defined
This weekend I made a Cherry Cobbler using the classic Southern recipe of a cup of sugar, a cup of self-rising floor, a cup of milk, a stick of butter, and fruit. (Melt the butter in a 9x13 pan, mix the sugar, milk and flour and pour in, add the fruit, bake. Easy peasy.) Kid Flash was a bit taken aback at how thin my batter was, and I didn't really understand that until later, when we were eating it. He remarked that it was more cake-like than he was expecting, and that to him cobbler suggested something with a more crumbly-crumb type topping. We discussed it a bit and figured out that he meant another recipe entirely, cold butter creamed with sugar and flour and spread over fruit, then baked. To me that is a Crumble, or a Crisp, not a cobbler. I also noted that if there had been oatmeal in the topping it would have been a "Brown Betty." So... is this a regional difference? What does cobbler evoke to you?
This has got to be regional, right? To me, you have the correct definitions (I expect a cobbler to be fruit suspended in an almost pancakey batter; a crisp is crunchy on top). The only other thing I would call a cobbler would be the fruit and sugar with biscuit dough dropped on the top.
(I should add that I feel like a traitor to the South, because I'll take a crisp over a cobbler any day.)
Don't know about the US, but in the UK the butter/flour/sugar top (sometimes including oats) is definitely a crumble which is quite traditional. Most times done with either apple, rhubarb, gooseberry or a mix of berries including blackberry, raspberries etc... The Swedish do a similar version, more oats in the topping and blueberry as the fruit. Tasty!
Wouldn't a biscuit topping make it a shortcake?