LH is operating yesterday's flight today at 6pm, so (2) fights are flying that route today, yesterday's, and today's. They don't need the 1st segment rebooked, they just show up for it today, but LH needs to rebook their connecting flight today at the airport if they haven't already done so.

I'm guessing your questiong to LH reservations didn't do much as it is still yesterday's flight they are on, and the details of that are being dealt with on an operational basis at the airport like the agents told them.

*edit* I see milepig posted the same 1 min b4 me.