If the first plant actually died, it is dead and no amount of watering will revive it. Perhaps there was another, smaller plant in the pot that responded to your watering.

Can you identify either of them?

They both look like variations of mints to me. Herbs usually develop better taste when they get a lot of sun and some heat. Have you tried this? Also, don't over-fertilize herbs; that can mean a loss of flavor too.

'Herbs usually develop better taste when they get a lot of sun and some heat.'

Mint is native to temperate climates and damp locations. It isn't a Mediterranean type herb, eg rosemary, sage, lavender etc, and it doesn't need heat or full sun. But it does need more light than is available indoors without grow lamps.

Both plants are clearly mints but there are hundreds of cultivars and fortuitous hybrids.

The mint is on my south facing balcony where it is used to receiving several hours of sun a day. Will the amount of water it gets influence the taste?

I don't think so. But since it needs the amount of water it needs there's nothing you can do to change that.

Same thing happening to me, except some of the old was still on the stem. Everything that grew in new is a different texture completely. I did fertilize/replant and its grown in crazy after partially dying.