Personally I would not try using the H5P plugin and the core H5P functionality at the same time, as I feel at some level, somewhere, they could interfere with each other.
So I would likely disable the plugin, and try using the core functionality.
However, if you prefer the plugin (without the overhead of the H5P core content bank), then uninstalling and reinstalling the H5P plugin might help.
As Beatriz says, the H5P plugin is maintained by Joubel, and the H5P core in Moodle is then adapted by Moodle HQ - I doubt anyone aside from Moodle HQ developers would be able to detail the 'differences' between the two, although in terms of the ways each content type works it should be identical.
I note you mentioned moving to your own
server.
One thing I might test is whether you can create this type of content on a
MoodleCloud site, with Boost Theme, and the same browser and connection you usually use.
If so, it may in fact be an issue with your server - speed - PHP - who knows!?