Happy cake day!

Personally I find Euclidean easy to remember because it matches the much more general Euclidean geometry. So you just remember "this is like, real maths". Manhattan distance is easy to remember because it does basically "refer to the metrics in terms of what they are", so long as you remember that Manhattan famously is a grid. Chebyshev is the hardest, but for me it's a simple matter of "the one that's left over".

I have no idea, based on the name, what diamond and square metrics are supposed to be.

Chebyshev distance can also be called chessboard distance if you want something more descriptive.

Fair point. I actually don't know what, if anything, the D&D (or Pathfinder) rules say on this matter. I've always just treated it as a natural 3D extension of the 2D grid rules. If they're three squares in one direction, same square in the other, and 10 feet up, I'd treat that as 15 feet away because of Chebyshev rules.

D&D's targeting rules are quite strange, but yes, it's very explicit that Chebyshev is used in 5e by default, if playing on a grid. On page 192 of the 5.0e PHB:

To enter a square, you must have at least 1 square of movement left, even if the square is diagonally adjacent to the square you're in.

The DMG presents, on page 252, an optional variant of the optional grid rules, which is to treat it the same as Pathfinder 2e does (alternating 5 ft and 10 ft):

The Player's Handbook presents a simple method for counting movement and measuring range on a grid: count every square as 5 feet, even if you're moving diagonally. ... This optional rule provides more realism.

When measuring range or moving diagonally on a grid, the first diagonal square counts as 5 feet, but the second one counts as 10 feet. This pattern...continues when you're counting diagonally even if you move horizontally or vertically between different bits of diagonal movement.

As for the value of cube vs sphere in the context of Chebyshev ranges, there are two key differences.

First, cubes measure side length, spheres measure radius. A 10 ft cube covers 4 squares. A 10 ft sphere covers 16.

Second, and more importantly (since the above could easily be translated by using only cubes or only spheres throughout the system, with either half or double the numbers), cubes are cast from one side, whereas spheres are cast from the centre. If you're standing in the front line with enemies in front of you and allies behind, a cube cast with you as its origin point will hit either allies only or enemies only, but not both. A sphere cast with you at its origin point will affect both allies and enemies. Note that the rules for cube, on page 204 of the 5.0 PHB say "A cube's point of origin is not included in the cube's area of effect, unless you decide otherwise." So you could include yourself and your allies, or you could include enemies but not yourself, if you so desired. Or, less likely, you could include allies but not yourself, or enemies and yourself.

From memory, cube spells are mostly cast from a range of "self", which is where this becomes an important distinction. If a spell has a range of X feet and cube, then the main difference is just that its area is smaller but its reach is longer than a sphere with the same numbers.

Fireball says radius, but in a non-Euclidian geometry radius doesn't translate to a Euclidian sphere. Embrace the cube of constant radius!

Thanks!