Definitely Alternator?

One thing first. The old bit, of running with the negative lead disconnected is a no no on newer cars. It can cause computer damage. Before I'd start spending hard earned money on parts I'd do a thorough cleaning of the battery posts and cable connectors with a wire brush, then have the battery tested on a load meter and then have the alternator checked out completely. It sounds like the alternator is not up to the load, either because it is flaky or is under sized. However bad connections or a battery with deteriorated plates could be the cause. Just don't waste money by throwing parts at it willy nilly as some have suggested.