Chris, for wings, I still find myself sometimes using different move-optimal commutators for the same pieces on different solves. So move-optimal doesn't always mean fixed. It probably should always mean fixed, but if you're bad at it, which I still am with wings, it might be somewhat freestyle. Or would you consider someone who does BH commutators freestyle to not be doing BH, simply because they're coming up with them on the fly?
(An example: FDr -> LDf -> FUr might be done [D', f U f'] or [F' u' F, D]. Both perfectly valid BH commutators.)
I really need to work on this; wings are holding me back so much on big cubes.
This is good, as this issue means we need to solidify what it means when a method is "fixed." I sometimes do this as well, and it can depend on how I am holding the cube at the time. For example, if the previous alg had me do a y' from the initial orientation I would definitely use the [D', f U f'] alg (or, since I'm already at a y' rotation from solved I would really be doing [D', r U r'] y).
If the previous alg had me do a y rotation from solved, then I would use the [F' u' F, D] alg (only, since I would already be at a y rotation from solved it would really be [L' u' L, D] y')
I would say that since both are move optimal, and you alternate between the two, that this method is still "fixed." Even if one was move optimal and the other was not, say D2 l U2 l' D l U2 l' D, as long as you alternate between those two it is still fixed.
If lots of algs have mixes between move optimal variations and non-move optimal variations, and you swap between the two on any given case, then I would argue that your method is becoming more of a "free, direct solve, 3-cycle method" and thus more "freestyle" than BH.
So for example, for the cycle you gave if I consistently alternated between the 9 move alg and the 8 move alg (and I did this on a number of other cycles as well), then that is more of a "free" solving approach in my opinion than a "fixed" solving approach.
As to you saying that "you're bad at wings," what makes you say that? If using an approach that's a bit more free makes more sense to you, then do that. If there's one thing I've learned, it's that free and fixed approaches seem to be a preference to the individual. I'm the kind of person that does better with fixed approach methods. Like Stefan, I think the "braindeadness" of a fixed approach is an advantage for ease of solving. For some people it seems that having a freer method allows for more innovation during the solve, which can lead to solves with better flow, and thus faster solves. Mike, from what you're saying it seems like your method is still fixed, or are you alternating a lot between different algs on many different cases?
I say "free, direct solving, 3-cycle" variations of BH are maybe not strictly BH. In my opinion they're a modified version, or variation of BH - simple as that. I personally use a "fixed, direct, 3-cycle" variation of BH - I do not use strictly BH myself. I often use 9 move algs with better flow in place of the move optimal 8 movers. I do this for all piece types, but I do it in a predictable way, and I don't alternate between the 9 mover with better flow and the 8 mover with less flow. I just use the 9 mover with better flow. So, this is still a "fixed" approach, in my opinion, but it's also still a variation of what could be defined as "strictly BH." That's what works for me, though.