So, I decided to pick this up yesterday and I came with an acceptable solution.
I reïnstalled my Photoshop CC 2019 CLEAN. That means I finally had the chance to start off with an English version. The Photoshop at work was configured to be in the Dutch language, but that’s just horrible. Just believe me on that.
The reason why I post this reply, half an year after the problem is that some other might have the same problem or I, myself in the future might reference to this post. I do that more often. Look up other problems that I encountered and asked for a solution for on this forum. I hope I might help someone out with this.
I started over. Here is the script that I made (based on the steps I want it to do, as seen in the first message of this post).

So let’s open up EVERY image that we want to convert with this script. Usually I open 5 to 10 (PSD) files.
Now I had the problem that the actions do not seem to be very variable in where to save it and how to name it, so as you can guess I ended up in the following menu (File > Automate > Batch):

This does exactly what I want (note: the save location has changed (from C:\ to E:\) due to the hard disk space being insufficient. So the script still contains C:\ and the automate thing saves it to E:\, you can change this at any time over here, without changing anything in your action script.
Problem:
It adds “_WEB” to everyfile name. So in this example our original file is called “PICTURE.PSD”. The output comes like the following four files.
– PICTURE_WEB.JPG
– PICTURE_WEB_WEB.EPS
– PICTURE_WEB_WEB_WEB.JPG
– PICTURE_WEB_WEB_WEB_WEB.PNG
The output is good! Exactly what I want. Only the file names are a bit messed up. So I manually change them to the following:
– PICTURE.JPG
– PICTURE.EPS
– PICTURE_WEB.JPG
– PICTURE_WEB.PNG
(I delete ‘_WEB’ once on the first file, twice on the second. Twice again on the third so one is left behind and three ‘_WEB’s on the last one.
I’m happy with this outcome. If anyone knows how to fix the last problem, that would be very welcome. For now I’m just really happy that I’m able to convert my files in one press of a button.
Thanks guys, for helping me out on this 🙂