There is likely a good technical explanation to be offered; I'm not feeling up to producing one. I do have some handwaving intuition that might help.

Your example is based in a finite language/signature (one binary operation symbol), so I will restrict myself to finite functional languages, even though some things might hold for countable languages. The subvariety lattice for a given finite signature corresponds in a dual fashion to the lattice of closed (first-order) equational theories in that signature, which in turn correspond to the lattice of certain (fully-invariant, I think) congruences in the free term algebra on countably many generators. This lattice can be pretty woolly, but you've found a nice sublattice of it which corresponds to the subvarieties of bands. Further, this nice sublattice has a property of representation that you pointed out: relative to the congruence that defines or describes all bands, any further congruence containing it is principally and singularly defined, needing just one pair of terms to be defined to generate the rest of it.

Why is this? In the case of bands, there is a simple enough set up: reduce all terms using associativity to a nice normal form (semigroup words), then use idempotency to further reduce possibilities. If I recall correctly, idempotency really cuts down on the available distinct words: in particular, bands are a locally finite variety, so for each n there will be only finitely many distinct terms in n variables to choose from.

Why do you get to use exactly one identity? This is less clear to me, but recall that an identity in n variables implies (by substitution) several identities in m variables for m less than n. There may be a way in the case of bands to order the relatively free terms by implicational strength, so that the result is a linear order. Then any collection of identites can be reduced by some process to one.

The last part involved a bit of arm waving as well. I want to emphasize the point that if you set up an initial base of identities in your theory, you can maneuver into a position where there aren't many more steps to x=y, or triviality. If you start high enough (in the lattice of fully invariant congruences of the term algebra), I think you can carve out many theories which represent their completions in such a nice manner.

There is likely a good technical explanation to be offered; I'm not feeling up to producing one. I do have some handwaving intuition that might help.

Your example is based in a finite language/signature (one binary operation symbol), so I will restrict myself to finite functional languages, even though some things might hold for countable languages. The subvariety lattice for a given finite signature corresponds in a dual fashion to the lattice of closed (first-order) equational theories in that signature, which in turn correspond to the lattice of certain (fully-invariant, I think) congruences in the free term algebra on countably many generators. This lattice can be pretty woolly, but you've found a nice sublattice of it which corresponds to the subvarieties of bands. Further, this nice sublattice has a property of representation that you pointed out: relative to the congruence that defines or describes all bands, any further congruence containing it is principally and singularly defined, needing just one pair of terms to be defined to generate the rest of it.

Why is this? In the case of bands, there is a simple enough set up: reduce all terms using associativity to a nice normal form (semigroup words), then use idempotency to further reduce possibilities. If I recall correctly, idempotency really cuts down on the available distinct words: in particular, bands are a locally finite variety, so for each n there will be only finitely many distinct terms in n variables to choose from.

Why do you get to use exactly one identity? This is less clear to me, but recall that an identity in n variables implies (by substitution) several identities in m variables for m less than n. There may be a way in the case of bands to order the relatively free terms by implicational strength, so that the result is a linear (or not very wide partial) order. Then any collection of identites can be reduced by some process to one.

The last part involved a bit of arm waving as well. I want to emphasize the point that if you set up an initial base of identities in your theory, you can maneuver into a position where there aren't many more steps to x=y, or triviality. If you start high enough (in the lattice of fully invariant congruences of the term algebra), I think you can carve out many theories which represent their completions in such a nice manner.