I thought it would be a good idea to organize the different families by complexity, in other words, how many states do you need to demonstrate this behaviour. I will require each example to have a designated halt state (even though they never reach it). This is because another goal of this thread is to test the strength of our current decider system.
1 state
Translated Cyclers
E.g. A1R 1R-
As there can only be one non-halt state in this case, the only non-halting possibility is for a translated cycler.
2 states
Cyclers
E.g. 1RB 0RB ; 0LA 1R-
There still seems to not be that much room for anything more than simple cycles with two states
These machines are indeed the only type of machines that occur in three states. A bilateral bouncer decider and an exponential counter decider are enough to settle BB(3).
I suggest a further distinction of complexity class, the rate of occurrence of different types of machines. This scale should agree with the original classification around how many states are needed to produce a behaviour. It will also allow for a finer distinction.
For example, of the 40 three state Turing machines which are not halting, cyclers, or translated cyclers, 37 are bouncers, while only 3 are counters. According to Brady (see above paper), this same trend continues for 4 state machines, with bouncers occurring more than 10 times as often as counters (unfortunately, Brady does not distinguish bilateral from unilateral bouncers in the paper).
EDIT: I was mistaken, unilateral bouncers do occur in three state machines. Therefore they would need to be included in the three state category.