Thank you for your fast answer!
I now know why an alternative player was displayed by jDownloads.
In my /media folder there was an folder called plg_sometechiemediaplayerbridgelite.
Don't know which component it put there but after a rename of the folder and a browser cache cleanup the HTML5 player apears!
So my problem seems to be solved, great!
Indeed a default layout for all browsers by jDownloads would be great (with maybe a switch between the default HTML5 player layout and the jDownloads player layout).
Could be a nice future request if development has some time left
Thanks for your support!