Number of 5-state TMs is 21^10 = 16,679,880,978,201; coming from (1 + writemovestate)^2*state; difference with your formula is that in our model, halting is encoded using undefined transition.
"essentially different" is not a statically-checked property. It is discovered by the enumeration algorithm (Tree Normal Form, see Section 3 of https://arxiv.org/pdf/2509.12337); in particular, for each machine, the algorithm needs to know it if will ever reach an undefined transition because if it does it will visit the children of that machine (i.e. all ways to set that reached undefined transition).
Knowing if an undefined transition will be ever reached is not computable, hence knowing the number of enumerated machines is not computable in general and is as hard as solving BB(5).
"essentially different" is not a statically-checked property. It is discovered by the enumeration algorithm (Tree Normal Form, see Section 3 of https://arxiv.org/pdf/2509.12337); in particular, for each machine, the algorithm needs to know it if will ever reach an undefined transition because if it does it will visit the children of that machine (i.e. all ways to set that reached undefined transition).
Knowing if an undefined transition will be ever reached is not computable, hence knowing the number of enumerated machines is not computable in general and is as hard as solving BB(5).