Thanks for looking at this. My thinking too was to equate dimensionality with the number of state variables.
Having said this, might path dependency complicate things? Prepayments, knock-ins/outs and the like?
I was thinking of dt as constant, sorry if not clear.
I suppose we can qualify things even more by saying that state variables are stochastic by definition and hence it excludes deterministic variables that arise in SDEs. For example, an Asian option would be 1 factor because the average term A has no diffusion. As a PDE(S,A) it looks like a 2d problem but not quite. Actually, solving Asian options numerically is more difficult than a fill 2-factor stochastic and where it is easy to burn one's fingers. Some challenges when taking 'standard' FDM are discussed here
Similarly, Cheyette would also be 1 factor according to this rule.
All these 'extra' constraints do not increase the dimensionality of the problem but the downstream numerics look complexer.
Anyways, that's my 2 cents.