The argument that they should be outside the circle of their general ability types seems to hinge on their different timing.
I'm not sure I'd agree with this assessment. The argument that "forced reactions" are initiated differently from "reactions" is not really about timing. It's about the fact that the rules talk about responses being initiated and forced reactions being resolved. As Horobod has demonstrated, that's a pretty consistent difference in terminology throughout the RRG - too consistent for them to be treated as synonymous (the way "triggered" and "initiated" often were for passive effects in 1.0, for example).
Pointing to the fact that the first player gets to determine the order in which forced reactions are resolved, but not getting to determine the order in which reactions are initiated, is also not really about timing. It's about consistency. Generally, the First Player chooses the order in which things resolve when they cannot resolve at the same time. The First Player never chooses the order in which players initiate abilities.
So, if forced reactions are initiated by players, but the first player decides the order in which they are initiated, we are looking at an exception to all other situations and rules involving first player choice. We are carving out the only situation in which the first player actually decides when things initiate. (This is also inconsistent with the RRG entries describing forced reactions as something that "resolve" rather than as reactions that are "initiated.")
But if all valid forced reactions are automatically initiated at the same time by the game (and not by the players), the first player deciding the order of their resolution is not only consistent with all other times when the first player privilege is applied, it is consistent with the RRG entries describing forced reactions in terms of their resolution.
This is why I say that, in the lack of specific text saying "forced effects are triggered by the game, not the player," the "by the game" interpretation is stronger than the "by the player" interpretation. It is more internally consistent with the other rules of the game.
But you say you have no doubt this interpretation is correct. Why do you think that? If the two interpretations were equally justified by the rules, wouldn't there be doubt for one interpretation over the other?