I see you've got the Galaxy Watch. Mine is the Watch 4 Classic. My charger that came with it, either plugs into a USB slot on a computer or you can plug the charger into the charging brick of your phone. If, of course, you have a samsung charger with a removable USB cable that is.
My charger doesn't have any lights on it. I've noticed though, that my watch does stop charging once 100% is reached. Looks like the only way you can top up to 100% again is to lift the watch off it, then place it back on. I take it that is some safety feature.
I get a notification on my phone once the watch has reached 100% charge. Does your phone do that?