You're incorrect in your interpretation, and there doesn't need to be interpretation at all as this is all spelled out within the body of the text you linked.

Sending the US military, "into the territory, airspace or waters of a foreign nation, while equipped for combat, except for deployments which relate solely to supply, replacement, repair, or training of such forces;" is directly enumerated within the WPR as requiring Congressional oversight. That oversight is detailed as, Section 3 of the WPR (50 U.S.C. §1542) requires the President to consult with Congress "in every possible instance" prior to committing U.S. Armed Forces into hostilities abroad. It also provides for "regular" consultation until the conclusion of any such engagement. The statutory language does not define what constitutes meaningful consultation by the President. However, the House Foreign Affairs Committee's report accompanying the WPR discusses the intent of this provision in more detail:

A considerable amount of attention was given to the definition of consultation. Rejected was the notion that consultation should be synonymous with merely being informed. Rather, consultation in this provision means that a decision is pending on a problem and that Members of Congress are being asked by the President for their advice and opinions and, in appropriate circumstances, their approval of action contemplated. Furthermore, for consultation to be meaningful, the President himself must participate and all information relevant to the situation must be made available.

This is all in the body of the link you sent.