Is this battery issue or screen?

NVRAM contents are preserved by battery power on portable Macs. Given the fact the battery is known to be dead it's not surprising that resetting NVRAM results in a successful boot.

It is not possible to provide a definitive answer, but the symptoms you describe and the actions you took suggest replacing the battery might be worthwhile:

Obtaining service for your Apple product after an expired warranty - Apple Support

The battery shouldn't have any bearing on the NVRAM except for being able to retain those settings when the charger is disconnected. A bad battery can make it more difficult or even impossible to power on the laptop since a bad or discharged battery can prevent the full power of the charger from making it to the rest of the system.

At first I thought you had an LCD Panel (Display Assembly) issue, but seeing the different screenshots and considering the PRAM Reset and the black screen issues, I would say you most likely have a bad Logic Board. You can try connecting an external display and enable mirroring in order to see whether the same screen issues appear on the external display as well. Otherwise, there really isn't an easy way of confirming the problem without testing with known good components. For such an old laptop where you already need a battery, I don't think it is economical to pursue any repairs.

pajczukin wrote:

But at the moment my mother use it and as I told she use it very rarely, about 2 hours in week or even less, and only for internet. And for such purposes I claim that old Macbook is still great computer.

There are much less expensive options for this scenario. One involves using just about any old used PC laptop and installing Linux Mint on it. As long as you get a laptop with a decent CPU and not one of those low end Celeron, Atom, or low end AMD e-series CPUs, then Linux should run fine on it. A used business class laptop is a good choice. Linux can work well for browsing the Internet with as little as 4GB of RAM and has access to the latest versions of the popular browsers (Firefox, Chrome, Vivaldi, and others), plus access to lots of free open source software in the Linux distribution's software repositories such as LibreOffice.

Another option is to purchase a Chromebook. A new Chromebook is likely going to be less expensive than repairing this old Apple laptop. Most are available for less than $300 US. Our organization utilizes both Dell & HP Chromebooks (both sturdy although I generally prefer Dell over HP in most cases), some other brands may be much less sturdy (Acer). I'm not personally familiar with any other Chromebook brands. If well cared for, then even the Acer would probably be Ok for such light use.