Destiny 2 – How Does Exotic Weapon Quest Scaling Work?

You described the differences pretty well.

In Destiny 1, shaders were normal items - they were not consumable. There was only one shader slot, and it applied to all your armor (weapons were not affected (related, most exotic weapons had one or two ornaments that worked similar to shaders, but were consumable)). You could change the shader at will, and the shader would not be destroyed. Later on in Destiny 1's development they added a shader kiosk where you could get a new copy of any shader you have ever found which would allow you to effectively use the same shader across all your characters.

In Destiny 2, shaders are now applied per item and can be applied to weapons, sparrows, ships, and ghosts in addition to armor. Shaders are also consumable - once you've applied a shader, you can't use it again. If you replace it with a different shader, the first shader is gone. When a shader drops, you typically get a few copies of it (I've gotten 3 at once, not sure if that is guaranteed).

The main issue some members of the community have with the new system is that in addition to being consumable (which means you tend to think about applying a shader, since you can't use it again), is that you can buy them with real money via the Eververse store. The conspiracy theory is that shaders were made consumable so that Bungie would make more money on microtransactions.

Kinetic weapons have their Power value in white, with no elemental symbol (red flame, blue spark, purple vortex) by the value.

Energy weapons and Power weapons both have elemental damage. Those are the symbols I mentioned above and correspond to Solar, Arc, and Void.

Exotics notwithstanding (though I believe only Fighting Lion breaks the mold by being a Grenade Launcher in your Energy Weapon slot),

Kinetic weapons and Energy weapons share the same weapon type. These are:

Auto Rifles, Pulse Rifles, Scout Rifles, Handcannons, Sidearms, Sub-machineguns

Power weapons are exclusively:

Grenade Launchers, Rocket Launchers, Fusion Rifles, Sniper Rifles, Swords, Shotguns, and Linear Fusion Rifles.

And as you have discovered, you can always press Q or LT, whichever your appropriate keybinding is, to preview the weapon compared to your current equipped weapon in that slot.