Did you check voltages with the multimeter ? What's the battery voltage measured across the posts with the battery disconnected, connected and during attempted starting ? If it doesn't fall below 12-13V then it's unlikely the battery is the problem and you need to check voltages from each of the posts to the opposite pole in the cars wiring, i.e stick one test lead on the -ve post and probe around the positive wiring from the connector on the battery to the fuse plate attached to the battery using the other test lead. See if the voltages are above 12V. If they are then do the same by checking from the battery +ve post to various points on the -ve wiring including the bodywork, engine etc. If the battery isn't faulty there must be a poor connection somewhere in the wiring, just need to find it. Given that waggling the connections to the battery seems to help, it seems likely the problem is with these wires somewhere.