Koolminx:
I think I understand your present confusion: Knowing new words/terms without knowing what they refer to in the real world is just confusing. You need to put some actual images to those terms. Checking out casting videos is better, and getting actual instruction is better yet. But I'll give you a lightning survey now, to begin to make sense of it.
Almost all casting with single-hand fly rods is as you describe it: it's called "overhead casting" from your master arm-side. 'Nuff said. When a strong wind is blowing against your master arm, you can make the same moves with your arm pointed across your body, so that your hand and arm, and the fly line, are moving back and forth on your downwind side. You can cast to moderate distance that way without getting your fly up your nostril.
The one other cast that other old timers all knew and used when their backcast space was full of leaves and branches, is the roll cast. It's best practiced on still or slow-moving water: With the line on the water in front of you and your rod low and pointing outward, toward where you want to cast, raise the rod at medium speed, nearly vertical, but just clearing your strong side. Pause and let the line sag down beside you. Cast forward, stopping the tip at about 45 degrees. The line and fly will go where the rod's pointing. With practice, you can roll cast 30 - 40 feet.
Now to spey casts, and the new terminology. Spey casts are water-anchored; that is, the fly and leader stay on the water while you're loading your rod (either a single-handed rod or a spey rod) with energy for the forward cast. Water-anchoring makes loading the rod easier to do. (If that sounds like the roll cast I just described, you're right. The roll cast is a sort of low-energy spey cast, without a lateral change of direction.)
I'm going to jump out of place here, and describe the second half of all spey casts, because that second half is the same for all the spey casts. It's called a D-loop back cast, because to an observer standing to one side, the backcast momentarily looks like a capital D. The purpose of the first half of all spey casts is to bring your fly from downstream, where it ended up at the conclusion of your previous cast and presentation (that's called "the dangle") back up toward you. Ideally, your fly, halfway through a spey cast, is on the water one to 1 1/2 rod length's in front of you, usually on your strong side, the side where you'll be gripping the upper end of a spey rod's grip. Anyway, fling a loop of line into the air behind your strong side without pulling the fly out of the water. When the line is momentarily forming the D-loop in the air behind you, deliberately stroke the rod forward. Stop the rod tip high; aim for the horizon, not the water. You just made a spey cast!
Now let's back up to the beginning and look at a few spey casts. I use the double spey more than all other spey casts combined. It's easy, predictable and low-stress, kind of like doing tai chi. With the rod (either kind) on your downstream side, with the fly line and fly downstream at the dangle, raise the rod and sweep it across your front like a slow-moving windshield wiper until the rod is low and pointing upstream. The fly should be slightly downstream of you, at what's called the "anchor point." Now swing the rod in an arc downstream, about eye-height; continue the swing into a D-loop behind you. Cast it forward, over the water.
This is getting long, so I'll describe one other spey cast: the Snap-T. This is one of the newer spey casts, and looks like a stunt, but it's really an easy, practical spey cast. With the fly downstream at the dangle, face downstream and hold your rod in the side toward the stream. Starting with your rod low and pointing down the like toward your fly, raise the rod tip to about 60 degrees, then snap it down toward the water. The line and fly will jump out of the water and land slightly upstream of you. While that's happening, swing around and face the water. Immediately throw a D-loop behind you, then stroke it forward. Damn, this is fun!
To repeat: Spey casts can be made with spey rods, switch rods, and single-handed rods. They're very useful additions to the usual overhead casts with a single-hander. They have the same advantages: easy on the angler, because they avoid false-casting; good when dealing with obstructed space behind the caster. They're not the best way to air-dry a dry fly.
You'll have to learn to make spey casts with either hand/from either side. That's not as hard as it sounds. Or you can make spey casts across your body (sometimes called "kack-handed" by the British) to deal with the wind. When you learn to make a few spey casts from either side, you're ready to fish anywhere.