Are there different kinds of migraine?

Yes: There are several named migraine types, but the experience of migraine can be different for each patient even within the "same type". There are other headache syndromes, like cluster, that can be difficult to distinguish from migraines. Most of the differences are in features other than the headache, and won't matter as much to the sufferer, but can be important for providers to choose therapies.

Too many: Migraine is a collection of symptoms. The manner of expression of these symptoms can be very complex - changing even for a single patient over different attacks. The symptoms change with age. Autonomic symptoms can be very complicated with vomiting, diarrhea, sweating, flushing, pale, etc. Vascular changes including loss of vision, numbness, weakness, fainting, vertigo, stroke-like, etc.