I cloned the repo, then
$ cd universal_robot/ur_description/urdf
$ xacro --inorder u10_robot.urdf.xacro > ur10_original.urdf
(I have a feeling that is not the way that it should be done, but does the trick. Probably I should use ROS’s package environment variable or something like that.)
For converting the files, I used the first online tool I found: meshconvert.com. I know it’s not an ideal solution, I will definitely will check MeshLab!
I assume that this converter hacked the coordinate frames which caused my issue.
(I will check later your other responses too!)