The categories and sub categories of rods makes for interesting discussion but all too often this falls into the category of the fellow who walks into the fly shop picks a rod off the rack, balances it on his finger, wiggles it from side to side and up and down a bit, then presses it to the rug to look at the bend.
The expression on his face tells volumes as it's a bit of quizzical, a hint of a frown at times, and then a smile as he pronounces, "this is a fine fly rod indeed!" At times he may even get a couple of the others who hang out at the shop, to pick up rods, usually their favorites, go through the wiggle tests and such, and then launch into a soliloquy on the finer points of fly rods and why the IM-7s are better than the IM6s, etc.
In the end analysis it's your rod, and has to be the rod that works for you if you are to be happy.
The many terms and more that we use to describe rods, unfortunately means different things to different folks. Even a master rod builder who has been into what makes a rod tick may or may not be using the terms the same as you understand them.
Rod lightness, hardware, taper, modulous, density, slenderness, rod materials, and more go into making a rod tick to our liking. Fast does equate to stiff but also equates to taper, materials, hardware (especialy hardware weight) and so it goes. Orvis has a flex index that helps for sure to narrow down rods you might like if you know what rods you really like and how they equate to the Flex Index.
By far the best thing to do when contemplating a new rod is head to a shop, or gather a dozen fly fishing friends and start trying as many of the various rods as you can.
Lawn casting may not be the enth degree as far as test driving a rod goes but it will give you a great feel for the rod above and beyond the inshop wiggle and pontificate tests!
Above all be sure to get a rod that has bends in a parabolic curve!