Cannot author styles and paraformat in Chrome/Firefox
At our project, we are using AEM 6.2.
We've created a number of pages, and so far there were no problems.
However, myself and a colleague now want to start to do some authoring.
This is where we are encountering an issue:
1. Log in to AEM
2. Select a page and go into edit mode
3. Press on a text
4. Press the pencil icon (Edit)

5. The text editor is now opened, with a list of buttons in the top

When I press the Styles or Paraformat icon (highlighted in red), it really quickly opens it open (for Paraformat it shows paragraph, header 1, header 2) but then immediately closes the text editor and it goes back to the page itself in Edit mode.
I've used Chrome and Firefox (latest versions), in both cases it doesn't work. There are no console errors.
I've cleared cache and cookies, used incognito mode, not helping.
In Internet Explorer 11 it does work fine.
Here the behaviour is different as well.
1. Log in to AEM
2. Select a page and go into edit mode
3. Press on a text
4. Press the pencil icon (Edit)
5. It now shows only a few options, I have to press the enlarge button to show the text editor

6. The Styles and Paraformat buttons work fine now.

For the developers it works fine in all browsers. They are using identical browser versions, settings and the AEM same account.
Anyone else experienced this?