I've never in my life seen a Haflinger that looked anything like that, though people do tend to call anything short and stocky a "halflinger cross" if they're not sure (it sounds like the previous owners weren't!).

Pretty much only Exmoors look like Exmoors! Some of them are indeed microchipped, but some aren't; my pony is, but his sister never was, it's entirely up to the owners to do it. The 'hooded eye' is also called a 'toad eye' in England; it's that almost puffy-looking bit over the eyes. They've developed it to help shed rain and snow on the moors. If she gets a 'fan tail' in the wintertime (short, stiff hairs at the top of the tail that spread out like a fan, also to shed snow and water), it's definite she's an Exmoor, and probably full-bred.

I don't recognize that particular pony, but I bet someone in the Exmoor group here in the States or Canada would. There are only 40 -50here, so there are only so many options she could be!