Thinking of this tag usage, I'd say that 'Process Engineering' would be used when one tries to improve a process that already exists or would need to define something that's not structured yet.
Going a step back, quoting 'Engineering definition' from Google (apologies, as a non-native English speaker I like to double check I'm not - at least - completely missing the point. Just a bit is ok :) ):
a field of study or activity concerned with modification or
development in a particular area. "software engineering"
So, in this sense, 'process engineering' could make sense from a PM perspective as a tag to gather tickets related to process problems.
Being pragmatic, however, I believe the tag doesn't add much value as it stands. Some of the tagged postings could be tagged as methodology, like THIS and THIS.
All in all, I support J's answer of going question by question and analyzing / suggesting tag changes, but I believe that 'process engineering' tag would make sense from a theoretical point of view for PMSE.
As a side note, it seems that process and methodology-related tags are being used without much 'care'. Maybe would worth to highlight the diffs between them on the tags Excerptions?