How to prove the authenticity of a screenshot?

You cannot prove that. They were on your PC, fully under your control for some time. You could have tampered with them. Therefore you cannot prove that you did not tamper with them.

If you need to set up a legally safe solution then look for an independent third party and a way to have them store information in such a way that you can only trigger a store or read (e.g., a screenshot on a Citrix server to a write-once location).

You can never prove authenticity in a screenshot.

It is incredibly simple to change the apparent content of anything posted in a public place & needs no great skill to totally change the meaning into anything you like…

This fake screenshot took about 30 seconds in photoshop.

For those who didn't like my quick mockup the first time, here's another, lined up correctly… I chose to use the current version of the question rather then re-mock the original - the result would be the same either way.

Of course there is no sure shot way to find image manipulation attempts. But there are some basic techniques people use to manipulate images.

For example, people use photoshop clone tool to duplicate patterns/colors. It may be difficult to detect by manual observation but there are some tools to do that.

Have a look. The tool has a number of features to detect image manipulation. http://29a.ch/photo-forensics/#thumbnail-analysis