Rigify offset problem
Like in the video example, you can correct the offset by fixing the rotation of (in this case) the upper arms after animation is done, with new keyframes. Or you can prevent the problem by setting the rotation axis to X manual. There might be pros and cons with both ways. Illustrations below. You can right click on the images and choose to view in new tab to view large versions.
Compare the upper arm positions and roll of the metarig.
To get a better understanding of the rolls and rotaion of bones, you can turn on Axes, you'll find that under Armature icon, Object Data Properties, Viewport Display, check the box Axes. When a bone is selected you will be able to se the individual axes, (if you turn on the Wireframe or X-ray, you can see that the Y axis runs along inside the bone from bottom to top).
The upper arm of the generated Rigify, on the ORG layer.
Notice that the roll of the metarig upper arm and rigigy upper arm differ.
The upper arm of the generated Rigify, on the DEF layer.
Same goes for DEF-layer
Change settings for the bone
You can change the settings for a bone before you generate a rig. In this case the upper arm of the metarig is selected and under Rigify Type, you can set the Rotation Axis to X manual (instead of Automatic).
Now you can Generate or Re-Generate
Rigify upper arm roll after new settings.
The rigify rig upper arm is now similar to the metarig.