That second link says it - no battery. Strangely, Mactracker, which I nearly always recommend as one of the best sources of info on Macs, doesn't even mention the IIc. IIcx and IIci, but not IIc. I wonder if the IIc was produced for a specific and quite small market.
To the OP - I've asked for your other IIc thread to be moved here, where it's more likely to get attention than in Peripherals. You may find the two threads merged eventually.