Tap water does contains some minerals and salts but they are in dissolved form. so wiping the screen off before the water dries is the best way to go. you will see residue only if water stagnates and dries off on its own in hours. then you get those concentric whitish circles at the edges of the drop or spot of water.
The TS got those spots since the cloth used was too wet (probably soaking wet) instead of being damp. The cloth should be squeezed so that there;s no water left in it before wiping the monitor.
Also please avoid using distilled water from battery shops. A battery shop owner said they dilute a very small anount of sulphuric acid in it to prevent formation of dissolved oxygen in it.
I have a can of that distilled water in my home for my inverter battery. It dose smell like there's something added to it. It wont burn your skin, but you can feel that its not normal water by the smell.