Well, I think in that point we have rather opposing opinions.
IMHO Grey never and likely would never "want" war. IOTL to me he seemed VERY eager to avoid war if ever possible as long as what he saw as Britains interests were achieved otherways. And following Lord Palmerston : "We have no eternal allies, and we have no perpetual enemies. Our interests are eternal and perpetual, and those interests it is our duty to follow." this did not include to stand aside allying withFrance may come what wants.
On several occasions during the late July-crisis he tried to make it clear to Cambon and thereby France also ordering Sir Bertie to tell the same in Paris that France can and should not count on Britains military assistence/participation in a continental, military conflict. Grey tried to keep Germany and France away from any warring against each other at all by "threatening" Germany with Britains - potential - participation in war and "threatening" France with its - potential - NON-participation in war.
To me Grey was "pro-intervention" but not "pro-war" seeing intervention as a diplomatic one. Since 1904/05 despite whatever loud the 'sabe-rattling' was international conflicts were solved diplomatically :
both Moroccan crises, the 'Great Game' with Russia, the italo-türkish war, the Balkan-wars, Bagdad-Bahn and Lieman-von-Sanders affair.
All these crises had been solved without military involvement of a great or self-perceiving great power not counting the rather dilettantic colonial intervention against a dying, uncivilized wanna-be empire (italo-turkish war).
And Grey was - though unsuccessfully - trying to sell his 'conference-approach' on each and every occasion. Unfortunatly this time it did not stuck with the 'usual' participants, despite at one point or the other one or other diplomat of these ventilated a possible favour for this. ... if some conditions might be met beforehand.
So he 'failed' on his own approach and standards : the diplomatic one.
If he would have been really commited to "war-on-France-side-whatever" IMHO he wouldn't have carried the cabinets decisions as it did on making Belgium the "casus belli"-Litmus-test, neglecting the sending/embarkation of the BEF, waiting until midnight 4th when the germans finally rejected (by non-answering) even the last request/ultimatum to keep away from Belgium (and thereby the belgian channelcoast). ... even after everybody knew they had already invaded it.
From all written I've read he was truely and deeply affected when war actually came though he seems to have accepted it as the very last remaining card to be drawn.
(proposal for different development ot follow ... as soon as RL lets me)
Thank you and keep up with the next part, I look forward to it. My opinion likely solidified long ago under a different bias, but as I revisit different scenarios I welcome a different perspective to refresh how I see it. I might chalk up Grey's caution and approach to his sober understanding of the politics at home, but if I take him as not pro-war can I still see him as anti-German?