Automatic Weights not recalculating after moving Bone I am working with a fairly simple rig, as I'm pretty new to rigging/animation in general.

I created my bones, and later parented my mesh to the armature with automatic weights.

Later, I wasn't happy with some of the bone positions/sizes/orientations, so I moved them around in Edit mode.

I visited this question, in what I thought would be a simple way to recalculate the automatic weights. However, nothing appears to happen? Am I doing something incorrectly, or is there some type of problem keeping these weights from being regenerated.

In the provided gif, note that the weights stay the same after the bone is dragged way out of position.
I believe you should be in edit mode and not pose mode.

Suggestion. Manual weight painting is time costly.  If you have zero or or close to zero investment in manual weight painting then simply delete your existing [bone weight] work and start fresh to be time effective.  You have the luxury at the moment of discarding what might be a mistake rather than doing a sequence of recovery steps.

Simply assign automatic weights all over again.  That is 8 seconds of work. In your situation, the action of 

Parent Armature with automatic weights Control-P

can be done many times since you are not losing time costly manual weight painting.

Please know that a large bone next to a small bone will have generally more influence except where the small bone is extremely close to the mesh.