The system of ♢ ⧫ ⬠ prototiles might deserve research for its own merits, but it isn’t very convenient for encoding Penrose tilings. Conditions sufficient for Penrose-equivalence do not formalize as matching rules for it. They can be made a sort of local conditions on blocks of tiles, but these are not rules about edge-sharing of the type “you may connect this side to that side with specified orientation”, nor do restrictions on vertex figures help with it.
As for “fourth”, “fifth” etc. set of rules for Penrose tilings, Ī̲ found another (albeit closely related) system of three prototiles much more convenient for derivation of them and serves as a transition from ♢ ⧫ ⬠ to the P3 system.
For details, see the article Ī̲ wrote (but having difficulties to publish it even at arXiv.org due to their endorsement policy; see inside).