I've been inside most of the iPhones Apple manufactures although I have as yet to get under the hood of the new ”X”. Most of the problems I've found where video distortion is the issue can usually be found with the digitizer itself. Yes, it's possible to improperly reinstall the ribbon cable connector for the digitizer panel though it shouldn't have been necessary to do so in the first place for a simple battery replacement. I would recheck this connection for sure, but my next move would be a replacement digitizer!
Your problem my friend is almost certainly a bad display panel. There are a couple of ways to tackle this problem but the most efficient and easiest fix for this is to go over to eBay and pick up a decent used display. You'll find a good selection of them too. Plan on spending between $350 and $400 for the display assembly. The time to complete the swap will vary but you'll probably spend about an hour if it's your first time out.
I've been selling and repairing Macintosh computers for many years, and suggesting a backlight replacement as a fix for this problem is just way over the top! Generally speaking these display panels are fairly inexpensive... roughly $50 for a brand new bare panel, half that for a used part, and they're readily available. The time it takes to remove and replace the backlight alone exceeds the cost of a new panel altogether. However, the most efficient way to proceed with this particular repair is to purchase a complete used display assembly, replacing the damaged piece altogether. You're looking at spending 45 minutes to an hour, depending on your level of experience, to swap the complete display assembly, and with an expense of less then $50 you just can't beat the cost.
Don't back your machine up, and don't pay anything to anyone! Reset your phone as if you were starting it as a new phone. That should eliminate what's going on with it... and don't do any surfing with that phone!