I would argue that you're both wrong.

it was actually only a military conflict since it was never declared a war by Congress.

That's a terribly American-centric view of the war and of war in general. Had North Vietnam declared war on the United States would it not have constituted a war if Congress had failed to reciprocate? Revolutionary and civil wars (and the Vietnam war had elements of both) rarely involve formal "declarations of war."

Until then he holds that it was a full fledged war and not a conflict

War is a subset of conflict. The reason why there is a preference for the terminology of "conflict" is precisely because "war" has legalistic objections surrounding issues like, for example, the treatment of detainees.

It's for that reason that over the course of the 20th century you've seen a shift from the Third Geneva Convention in 1929, that is, officially, "Convention relative to the Treatment of Prisoners of War, Geneva July 27, 1929" to the universal adoption of the terminology of the Universal Law of Armed Conflict. The use of the terminology of "war" and "declarations of war" (which are a historical rarity over the past two centuries) was too easy for states to abuse, particularly after the experience of WWII where it was almost universally abused by all sides to one extent or another (the incomparable extreme of course being the holocaust.)

If you want an academic article citing discussion of this, you could start here, as an example: https://www.jstor.org/stable/1601299?seq=1#page_scan_tab_contents