No. It is not necessarily centered on the caster (though it can be).
A cube is not a burst power (to borrow a 4e term). The origination point of the cube is anywhere on a face, not necessarily the center. The term face when referring to a cube means the entire square comprising a side.
So you could feasibly center it on yourself, or place yourself in any other point of the cube's effect (and no, you don't have to attack yourself unless you want to).
I wanted to map this out real quick, just so we're all clear. Let's assume we have a 15' cube spell, and let's also use a grid just so we have a bit of clarity, and a limited set of points to operate from. There are 18 possible origin points for a 15' cube, representing two different 3x3 areas (the top and bottom face of a cube). For both of these the graph is the same, but the results are different. If you select the top face, the spell targets the 8 squares in your plane, and the 18 immediately below you. If you choose the origin as the bottom of the cube, then it targets the 8 in your plane and the 18 above you. Now which square is the origin:
XXX
XXX
XXX
Basically, you can be standing in any of the squares that are X and have the burst be those 9 squares. Let's look at a few more examples (Caster is C):
XCX
XXX
XXX
Here you step to a side and have it radiate to your left, right and two rows forward of you.
XXX
XCX
XXX
Here you center it on yourself.
CXX
XXX
XXX
Here you take a corner of the spell and cast it to your left and forward, and radiates out. It seems you should be able to be outside teh area too:
CXXX
XXX
XXX
Since you're adjacent (and heck that could be the middle of a face or the bottom or the top of it. (See the caveat below, this particular form falls much more into the "point" point of origin argument and may or may not be valid in all games.
CAVEAT: There are at least two camps developing around the term "point of origin" used in the text, and depending on which system you came from you may immediately interpret that differently. I have used in this response the version of the term influenced by my 4e experience, which infers that the point of origin is an entire square (this makes a ton of sense when you play with gridded combat as the default, which I do in my 5e games). However, there is a camp that comes from a different point of view that interprets "point of origin" as a singlular point (with an undefined diameter, likely just the part of the cast casting the spell...not sure, I'll let them speak for themselves). This POV would have the caster included in the spell if you cast in the ways that I describe above where the caster is fully within the Xs.
It's important to note that right now both of these interpretations are equally valid and subject to DM interpretation. I'll be using the "point is a whole space unless you don't want it to be" version, feel free to use whichever is most useful in your game (just be consistent about it).
What exactly constitutes the ground is left up to the GM. Everyone has a sense of what would constitue "the ground" at a given position. A GM ruling based on that will most likely satisfy anyone if made in advance (so no "your casting fails as that is not the ground"). I have never had problems with rulings like this, most players do not even question these as it is within their "sense of common" too. Even if a player contests it, they also have nothing to base that on and you trump them by being the GM.
The following is how I would rule:
- Being "at the height of the spell effect" is relative to "the ground".
- Casting is dependent on the ground at/under the target, not on the position of the caster.
- Something can only be "the ground" if a small or larger creature can stand on it for some time without balancing. So no throwing rocks up in the air and casting on them. Also no casting based on any tree you can climb with branches to stand on (unless it is the "World Tree" or something else on that level).
So in the masterfully illustrated example you cannot include the pterodactyl in the AoE while it flies around the tower at 75 feet. The ground is at the base of the tower in those positions. If he moves over the tower (to attack you, for example) you can include him in the cylinder as the tower is a stable platform you are able to stand on and thus is the ground there.
Best Answer
Yes, there is a difference
Or rather, No, your interpretation is incorrect.
First of all, let's look at the rules really quick.
Area of Effect on a Grid, DMG (I don't have the DMG in DnD Beyond, so this excerpt is from Xanathars)
So by the very nature of using a grid template, you're bound by the rules of choosing an intersection as the point of origin. Your character sits at the center of a square and occupies the entire square. The edges and corners of your square are 0 feet. The next square over, those far edges are all 5 feet away (including diagonals). Similarly, most characters have a melee reach of 5 feet which extends out towards the adjacent 8 squares. When creatures move out of those tiles, opportunity attacks are provoked. Range is always calculated from your tile boundary outwards and excludes the tile that the edges the range touches belong to. So 5 feet from your edge is another edge, but it excludes the next tile over. In this way, you can count the squares in a typical 5-foot grid system to calculate range.
So, a range of self is actually the "5-foot" range image you posted, since you are choosing a corner 0 feet away. If you had a 5-foot range, you could choose the intersection of any square up to 5 feet away, effectively moving the effect one tile away from yourself.
Remember that your point or origin is not always centered on a face of a cube, but rather anywhere on any face of that cube. So you must always pick a corner as the origin, but you could "translate" that cube any way you'd like.
Areas of Effect, Cube, PHB