I don't know if it was Luke Crane or Ron Edwards who coined "fail forward" - I think Edwards also called it "no whiffing".
But I don't think BW is Type A - or at least, I interpret Type A as "if you do it, you do it" and that is not BW. (HeroQuest Revised is also very similar to BW in it's explicit textual rejection of "if you do it, you do it".)
With the strong caveat that I haven't played BW only TB2 which isn't quite the same, I'm noting this rules text
You make tests during dramatic moments, when the outcome is uncertain.
When a player sets out a task for his character and states his intent, it is the GM’s job to inform him of the consequences of failure before the dice are rolled. “If you fail this…” should often be heard at the table. Let the players know the consequences of their actions.
Failure is not the end of the line, but it is complication that pushes the story in another direction. Once that is said, everyone knows what’s at stake and play can continue smoothly no matter what the result of the roll is.
We roll dice when a conflict arises.
When a test is failed, the GM introduces a complication.
"If nothing is at stake, say “yes” [to the player’s request], whatever they’redoing. Just go along with them. If they ask for information, give it tothem. If they have their characters go somewhere, they’re there. If theywant it, it’s theirs."*
Unless there is something at stake in the story you have created, don’t bother with the dice. Keep moving, keep describing, keep roleplaying. But as soon as a character wants something that he doesn’t have, needs to know something he doesn’t know, covets something that someone else has, roll the dice.
*The BW rules appear to incorporate this DotV rule.
Your documented play from last year shows some of this stuff in action
he did the only thing he could think of - as someone whose Circles include the Path of Spite, and who has a reputation as ill-favoured for himself and others, he looked to see if a bloodletting or surgical necromancer or similar ill-omened type might be nearby the scene!
But the Circles check failed: and so no friendly bloodletter appeared, but rather the Death Artist Thoth, who - for reasons not yet clear, but certainly not wholesome - carries a lock of the hair of Aedhros's dead spouse
Was Thoth in sight before the roll or added as a complication after? If the latter, then this Circles test looks like type A. But then
Today's session began with the Surgery attempt to treat Alicia's mortal wound. It failed badly (we have the notes somewhere, but from memory the Ob was 14 (7 doubled for no tools) and 1 success was achieved), and so her Health check to avoid acquiring the Mortally Wounded in the Head trait will be against Ob 19. A roll of the dice determined that it will take her 8 months to regain full consciousness, at which point she will then have to recover from a Traumatic Wound.
Was "8 months to regain full consciousness" in sight before the roll? Possibly this example follows a common pattern for type B. Which is, rather than exhaustively and inefficiently listing all scalable and forking consequences to meet the standard of informing player of the consequences of failure before the roll, GM indicates only their form or direction (e.g. "this might make it worse")... which here was likely clearly enough known to all.
So far, one might
possibly interpret the rules text "When a player sets out a task for his character and states his intent" to mean what it says: GM indicates consequences up front (i.e. it's type B) only when players set out tasks for their characters. Whereas for other tests only "When a test is failed, the GM introduces a complication" applies (making it type A). Said other tests being triggered not by player intent, but by dramatic moments and conflict. Or one could understand the complication as simply being the concrete form given to the consequences indicated up front.
Here's another example of a test from that same session, triggered I assume by player forcing matters along (intent)
Thoth, who has an Instinct to Always collect bits and pieces, also couldn't help but look around for Surgery tools, and succeeded on the Ob 10 test (Perception rolled for Beginner's Luck). A die of fate roll indicated that one corpse was available for collection, and Aedhros helped Thoth carry it off.
That's followed by something that perhaps was entertained as a dramatic moment the situation was ripe for
At one point, something particularly grotesque happened (I think, as narrated by my friend, a bit of the body Aedhros was holding onto sloughed off) and a Steel test was called for and failed. Hesitating, Aedhros dropped the body, and it nearly landed in the water - but Thoth, driven by his desire for corpses!, succeeded on a Power test to not let go.
The Steel test here could count as an example of one of those "other tests" prompted by a dramatic moment: so type A (the body nearly fell in the water). In any case, I expect in play that the shift between types is fluid, and the rules wordings seem open to interpreting as
if we can see consequences, then we are going to roll dice, and then we are going to see consequences (type B/A)
But those same rules can be interpreted as stating when to engage each type
when it's a dramatic moment or conflict, we roll dice, and then we are going to see consequences (type A)
when player states their intent and its a task, if we can see consequences, then we are going to roll dice (type B)
And sometimes these cases will overlap, leading to the merged type B/A form just above. I think what BW is going for is sensitivity to the ripeness of the situation and to the desire of players to force matters to a head. The former has the capacity to be wide in scope - to admit all sorts of possible additions to the fiction that are consistent with theme - while the latter has a tendency to be narrow in scope - urging tight extrapolation from present situation. The Circles test turns up Thoth. The Surgery test worsens or does not ameliorate Alicia's trauma.
So this quality - thematic scope - interacts with sequence (type A or B) to yield a gamut of types. Potentially letting type B/A above be reunderstood as
if the situation is ripe with wide thematic scope, then if we roll dice, we are going to see consequences which may be unexpected (type A-wide, may turn up Thoth)
if player forces things to a head in a way with narrow thematic scope, then if we can see consequences (their form is indicated up front), we are going to roll dice (type B-narrow, may protract Alicia's convalence)
And these can be paired in the opposite way, producing two more sub-types. One wouldn't in any case suppose that the type A / type B binary could possibly fully capture the subtlety and fluidity of actual play. It's more a conceptually useful simplification to understand and be intentional about consequences resolution.