One of those two, but more commonly when the other guys ran away

I'm not exactly sure how to answer this in any great depth but perhaps /u/DonaldFDraper can strike it from a different angle. This wasn't some ritualized "match" played between two football teams where they shook hands at the end and called it quits when one team had more points after 90 minutes. People went into battles hoping to achieve something and if that something was achieved the battle was won and if it wasn't it was anything from lost to sorta won.

It's just a giant game of "it depends"; some battles were won when a bridge was taken and the other side withdrew; such as at the Battle of Lodi. It was certainly a victory even though it was by no means decisive.

The answer to "when did they know a battle was won" is "when the other sides will to continue fighting was broken" be that through a strategic withdrawal, total breaking of the lines into route, or surrender en masse. It's frankly too varied of a situation to say "a battle was only over in this 20 year span when a general officially surrendered to the other"; it was hardly ever that tidy. I mean look at Leipzig which was one of Napoleon's few defeats; Napoleon lost "only" about 25% of his force and he never 'officially' surrendered to the winning commanding officer nor was his entire army captured or killed. He just withdrew from the field of battle. Just think about this logically; how did an army know a battle in any other war was over that you're more familiar with, say, WWII? When the other side cedes the field. It doesn't require a ceremony or total destruction.

Interestingly in an episode in Japanese history...

After his defeat on the plains of Ueda by Murakami Yoshikio, Takeda Harunobu (Shingen) is recorded to have remained encamped on the battlefield for 20 days, long after Murakami withdrew his own forces, before withdrawing back to Kai. No one's sure why he did that.

One of the theories was to be able to claim victory. One commonly used criteria to define the victor was who had control of the battlefield afterwards. And the Takeda by remaining there till after Murakami withdrew certainly did.

But just to demonstrate that this criteria was by any means set in stone, if that was Takeda's aim for staying on the field, no one bought it. Though the battle was more or less a tactical draw, the Takeda no longer had the morale, momentum, and organization (they lost many important people in the battle) to continue their campaign in Shinano and pulled back to Kai.