This answer was written assuming the device (phone) required 1.2A and that the charger was insufficient. As @Passerby rightfully points out, phones typically can charge on much less current. That said, while the typical 5V USB (500mA) might be adequate for charging, it is not necessarily sufficient for operating it with screen, backlight, etc. plus charging. I was not able to find detailed specifications for the phone.
Ripple may be the issue, but if you are still having touchscreen problems with a clean power supply, connect the phone to a solid bench supply and measure what it uses when you've got the screen on, and the battery needs charging. At least then you then will know whether a 1A supply is okay.
I don't think you understand current/amperage as it pertains to devices and power supplies.
Your phone is rated for 5V 1.2A, meaning it requires 5V 1.2A, not that it can handle it.
The "bad" power supply is underrated for the device: it can only supply 1A but the device needs 1.2A. You're not supplying enough current, and because the load is demanding more current than the power supply can deliver, unusual behavior ensues.
It's very possible you've damaged the phone. Stop using the underrated supply and get one that is 5V and at least 1.2A. Once you power it with the proper supply, then you can determine if the phone is permanently damaged.
You mentioned in comments that another supply with the same rating (5V 1A) appears to work.
Not all chargers are built the same. Some are capable of supplying more current than they are rated for, but only for a limited amount of time (overcurrent capacity). Depending on the quality of the supply, it may have overcurrent protection, thermal protection, etc. While it's true that noise/ripple may also be a factor, I would be far more suspicious of too little power. If you continue to use the underrated supply that appears to work, you might be okay, but you also might find the supply getting hot and eventually dying altogether.
It's possible the manufacturer of your second supply, Apple, has actually under-rated the supply to err on the side of caution. If you built a bridge to handle no more than exactly 100 tons, would you post a sign saying "Capacity: 100 tons?" No. Well-built power supplies will generally do the same.
Think of it this way, your supply that says 1A might be able to supply a bit more, but it's designed for what it is labeled for. Heatsinks, capacitors, bridge rectifiers, etc. were chosen to dissipate a given amount of heat for a given amount of output power.
Here's one more example: Say an LED's datasheet indicates that its forward current is 20mA. That's for a 100% duty cycle, meaning, if you power the LED up, you can give it 20mA continuously and it will operate at the specified conditions for approximately the expected lifetime. The datasheet will also usually provide some higher current ratings and a corresponding duty cycle, i.e: "80mA at 20%" This means you can drive the LED with four times the current, but only if its on-to-off ratio is 20% to 80%.
A power supply isn't like an LED, but the idea is that you can exceed specs on certain devices sometimes. A good power supply will often show derating curves, which are specifications as how the supply behaves in abnormal conditions. Typically this is when the supply is hotter than it is rated for, but it can also apply to things such as supply undervoltage and load overcurrent, etc.
Your Apple USB charger is basically a smaller, simpler version of the power supplies I am referring to. It's not going to have these features and lengthy datasheets showing you its characteristics.
I'd recommend getting a power supply that is rated for 5V 2A or something, then you know it's going to be more than adequate for the load. For about $13 you could pick up a Meanwell RS-15-5 which is 5V 3A, and it will be able to charge two such phones at the same time.
You could check out the ripple on your supplies with an oscilloscope, but whatever your findings, I would not keep using a supply that's not rated for the load.
$13 is a lot less money than buying a new phone, or cleaning up after a fire.