First, thanks to everyone for their advice/information.

As one poster noted, the warranty does state that Miller needs to be
notified of problems in writing and only then will Miller decide what is
to be done with the welder to get it serviced under warranty. Well, I
contacted Miller via email, and the reply stated that taking it to the
authorized service agent was the correct thing to do.

Weird. The warranty states in writing one thing, and emailing a warranty
request to Miller gives you a different answer. My guess is that it is
standard practice to simply take it to the shop, and the requirement for a
written notification to Miller may be there to create a loophole that
Miller could use at its discretion to deny warranty claims. (too cynical?)

Another comment in the email reply surprised me. Miller's stand regarding
warranty service is that any shop that did not sell the unit is "not
technically obligated to service it." This information was not given as a
reply to a specific question or comment, but rather added by the author of
the email when the service record for my welder was checked, and it was
shown that the welder was purchased from a dealer in one state, and is
currently being serviced in another.

So, it would be perfectly acceptable, per Miller's policy, for an
authorized service agent to flat-out refuse warranty service to Miller
equipment purchased elsewhere.

Had I known this before ordering the unit, two things would have been
different: I would not have even considered buying a Miller welder online,
and I would have leaned away from even getting a Miller, looking into
getting a Lincoln/Hobart/ESAB instead. (I would have favored the other
manufacturers even more heavily had I known the Miller was going to die
after about 20 minutes of use.)

The reply from Miller also leads me to believe that the discussion I had
with the counter person and manager about why I purchased it elsewhere may
have helped my cause somewhat. The service record for the welder
apparently shows the shop had ordered a part yesterday, so they didn't
wait a week to look at it after all. Either the discussion helped, or
the tech was extremely bored.