Is there a list of “types” of flintlock pistol used during the English civil war?

To answer the first part of your question, no there were no "brands" as such of the period, although gunsmiths might have stamped their mark on examples of particularly well crafted items. In the period, the production of firearms was initially limited to a relatively few expert craftsmen and artisans before the wars. However, the wars themselves demanded such large numbers of firearms, that anyone that could provide firearms of useable quality could find a buyer. In the case of the matchlock, this often required little expertise, but the flintlock mechanism needed rather more skill. The second part of your question is a little more difficult to answer. We look back on the development of firearms and tend to group the mechanism types together. However, in the period there were many different forms of firing mechanism which relied on a flint, and the design itself was in flux (as indeed was the musket itself, ranging from the lighter caliver to the heavier musket), with "types" such as the English lock, the French lock, Doglocks, Snaplock, and Snaphances all being recognisably a FORM of flintlock, but not the more universally accepted form of flintlock which graced the Charleville or the Brown Bess. Indeed, soldiers of the period seem to have referred to these by the collective term of 'firelocks' or the better know Snaphance, than by their particular designs.

As for 'rare' it depends what you mean. If by "classy but retro" probably a wheellock, but their complexity compared to the flintlock meant they were rapidly going out of use by the British Civil Wars although cavalry or officers might use them if they could not get flintlocks. I suspect you mean "an awesome example" probably a rifled doglock. Doglocks were common in England at the time, and capable (due to the 'dog') of sitting at half-cock. However, for military purposes rifling pistols was rarely used so it would be a private order made with the nobility in mind. I am unaware of any examples of these, as rifling of firearms (or 'screwed' firearms back then) was known but focused on shooting for game, and therefore on wealthy men's fowling pieces. These did see limited use in the wars, but as they were so small in number the examples are few. At the siege of Lathom House the Royalist Garrison made use of 'the best and choicest marksmen who usually attended the Earl [James Stanley, 7th Earl of Derby] in his hunting and other sports as huntsmen, keepers, fowlers and the like who continually kept watch with screwed guns and long fowling pieces to the great annoyance and loss of the enemy [Parliamentarian besiegers], especially of their commanders who were frequently killed in their trenches or as they came or went to or from them.'

According to Donald Reid there were some simple breech-loaders and even a repeating rifle, but these were made for nobility and royalty and all were musket-type rather than pistol.