A couple of weeks ago, we had a brief (non-fires) exchange re: a quote from JP 3-07.1 (JTTP FID), here and here.

The quote from the manual was this (my concern focused on the last sentence, bolded by me):

(3) In all cases, the strategic initiative and responsibility lie with the HN. To preserve its legitimacy and ensure a lasting solution to the problem, the host government must bear this responsibility. A decision for US forces to take the strategic initiative amounts to a transition to war.
You advised:

You should be concerned with the legal aspects because that is the intent of the paragraph you quote.
and that I have done - thought about it.

But, first, another point you raised (which I should have answered) - the dictionary and legal definition of "war" (the dictionary from you, with JMM bolding):

war (wôr)
n.
1.
a. A state of open, armed, often prolonged conflict carried on between nations, states, or parties...
That is the basic Geneva definition (an "armed conflict"), which expanded the Hague definition of "war" - slightly with respect to state on state conflicts; but substantially by bringing in non-state parties ("Powers") to armed conflicts via Common Articles 2 and 3.

Moving back to FID (Foreign Internal Defense, where "internal" and "defense" are key words), that can occur in the context with no armed conflict or in the context of an armed conflict (subject to Geneva). Both situations are covered in JP 3-07.1.

Far be from me to interpret definitively what the CJCS meant by "A decision for US forces to take the strategic initiative amounts to a transition to war", but here is what I think.

Where the US "takes charge" (takes the "strategic initiative"), we transition our status from being an "assister" to being a "belligerent". In the case of Iraq, a co-belligerent with them against their insugencies, I suppose. That would be a tricky position legally, given the executive agreement (whether a SOFA or SA is not particularly material) in effect. As you pointed out, 50 USC 33 (War Powers Resolution) could well come into play.

Astan is different. There, we are engaged in FID and assisting them (usually taking the primary role in personnel and materiel) in their COIN efforts. Add the UN-NATO mandate to that, which is peace enforcement (not peacekeeping). But, we are also carrying on a separate war (armed conflict) against AQ-Taliban pursuant to the 2001 AUMF, as a belligerent - which spills over into Pstan and, as to which, Astan is not a co-belligerent. That is a complex situation legally, and from all appearances militarily.

Vietnam was different still, but that is another thread.

The issues raised above are really a mix of legal and military, where the right and left hands have to co-ordinate via a common brain. In that respect, COL Newell seems a good choice for Iraq since he researched and wrote a very intelligent thesis on a related area where legal and military concerns have to be met. I've lauded that elsewhere.