In this thread Glitchy Touchscreen Caused by Charger the answer states it is from the USB charger not being rated high enough. I am using the Moto G 1st gen and upon using a charger rated for 5v 800 mah I experience the same issues. The only thing is that the Moto G is rated 500 mah to 1.5A. In the comments of this post i found a link to this How can electromagnetic interference from a USB charger cause a capacitive touchscreen to malfunction? . Which is the real solution, as when the charger is not plugged into the phone but into the wall nothing happens. One of the wierdest thing i have observed is that upon pluggin the phone into the cheap charger while holding it in one hand and using my laptop touchpad with my other hand it too wacks out until i stop holding the phone. I have also used another charger and it does work perfectly fine with that one, sort of strange. Hopefully you have an answer, thanks.
Chargers use the same frequencies in performing their voltage conversions as touch screens use to sense the user touching the screen, these frequencies are usually a in the range of a few hundred kHz. This means that the charger can interfere with the device.
Because most chargers are two pin devices the USB output is floating (i.e. not connected to ground). However there is often coupling to the high-frequency such that many volts of common-mode may appear on the output.
Since the user has capacitative coupling to ground this common mode voltage appears in series with the wanted signal used by the device for sensing.
You can often improve performance by touching the case of the phone or any exposed metal to ground the phone and reduce this interfering signal.
Most phones and tablets will attempt to change their frequency to try to avoid the interference but in some cases it may not be possible as the actual frequency that the charger uses will vary with load to optimize its efficiency, for example this will mean it changes as the battery charges or the phone changes what it is doing.
This does relate somewhat to whether the charger has a high enough output, but it is not guaranteed that a more powerful charger will be any better than a low power one.
As you have experienced the actual interference can vary greatly between different products and how the system is grounded, such as when you touched the laptop which was probably grounded through its power connection.
It can also vary with which wall outlet is used.