On a side note: there is a way to partially unify subspace optimum and local optimum by saying that the subspace optimum is a local optimum with respect to the local set of parameters you’re using to define the subspace. You’re at a local optimum with respect to defining the underlying space to optimize over (aka the subspace) and a local optimum within that space (the subspace). (Relatedly, moduli spaces.)
I like the distinction that you’re making and that you gave it a clear name.
Relatedly, there is the method of Lagrangian multipliers for solving things in the subspace.
On a side note: there is a way to partially unify subspace optimum and local optimum by saying that the subspace optimum is a local optimum with respect to the local set of parameters you’re using to define the subspace. You’re at a local optimum with respect to defining the underlying space to optimize over (aka the subspace) and a local optimum within that space (the subspace). (Relatedly, moduli spaces.)