Just to report back: At SVO, we went to the "Jazz" lounge. I had my Delta Platinum card in hand but no invitation. The lady let me and my two kids in, no questions asked. (Perfectly nice lounge, though I was surprised the hot water ran out three minutes into my shower.)

The flight to SVO had been from JFK. At JFK, I had seen the check-in agent filling out an invitation card; I had asked if I could take my children with me; he had stepped away to ask the head guy, come back, and written my name "+ 2 children." The card was for the KAL lounge at JFK.

(The KAL lounge was perfectly OK. When we went to board, we found the gate had been changed and the flight had been delayed for over an hour. Just out of curiosity, I walked us over to the AF lounge. The lady there said Sorry, the KAL lounge handles Aeroflot. I had no objection, though I had thought ST Elite Plus gets you into any ST lounge on an int'l flight.)