The chlorine in tap water is your enemy. Other general salts and crud in the water are fine.
The traces of chlorine will be gone if the glass was washed with tap water and dried (let's assume by dried, you mean, you poured the water out, and didn't just let it evaporate - see below), before using distilled or deionized water. It's good practice to rinse the bottle out with the DI or distilled water, just in general.
But it is not necessary for handling LSD.
well im no expert but there are differences between all those types of water:
-deionized is just that, without ions. meaning it can still contain salts or orther stable chemicals/minerals
No, it quite definitely doesn't have any salts in it, since by definition a salt is composed of ions!
when tap water dries out in a glass, it leaves behind all sorts of chemicals, actually everything that doesnt evaporate (as demonstrated by countless adverts showing washing machines being destroyed by water

).
so technically if you dry out a glass washed with tap water, and then pour into it distilled water, what you get is a glass of tap water.
to get a glass of pure distilled water you would have to rinse it thoroughly with distilled water first then let dry then pour distilled water into it.
To bolded part:
Only if you let the glass sit out until it dries. When most people talk about drying a glass, they mean that they pour the water out first, and may then store it upside down. Realistically, then, you only have a few drops of water left in the glass to evaporate. So it's like a few drops of tap water in a glass of DI water.
A quick google search indicated that the chlorine content of tap water is spec'ed at 4mg/L. In, say, 250ml glass, that's 1mg - clearly enough to shred a few doses of LSD. (assuming that it's 1:1 by mole, 1mg of chlorine would eat ~4mg of LSD)
If the glass is emptied out, and only a few drops (say, .5ml) is left, you'd have only 2 micrograms of chlorine there.
This all is, of course, ignoring the fact that chlorine is volatile, and most of it will evaporate with the water (while it's measured and reported on a Cl2 basis, it's present mostly as hypochlorite ions, which are what do the damage to LSD), so little if any will be left behind.
On another topic, the mechanism by which tap water can damage appliances like washing machines has nothing to do with water evaporating, and everything to do with the bizarre solubility behavior of carbonates (particularly calcium and magnesium carbonate), which are present as contaminants in "hard" water.