I agree with your points but want to add that higher wages don't lead to high inflation directly. Higher spending does. If you have high wages and high savings it wouldn't lead to inflation in the same way that high wages and high spending would.
If input costs like energy go up, which affect virtually every item, then that can lead to higher inflation because the aggregate spending across everything can't drop.
[EDIT] that is, yes, falling spending would have a "cooling" effect, but if costs for, say, productive land, go up fast enough, you can still see inflation in the face of falling spending.