Long story short, maybe. The FAA doesn't define "dual time" so it depends how you define it for your own purposes, but in practice most people use it to mean "training time". There's no definition of "dual" or "dual time" in 14 CFR part 1 or 61 and part 61 never mentions it, apart from a couple of references to "dual instruction". "Dual" is also not used in form 8710.
But "training received" is clearly defined, and 61.51 says you should log time in general if you need it to satisfy certification or currency requirements:
(a) Training time and aeronautical experience. Each person must
document and record the following time in a manner acceptable to the
(1) Training and aeronautical experience used to meet the requirements
for a certificate, rating, or flight review of this part.
(2) The aeronautical experience required for meeting the recent flight
experience requirements of this part.
So in your example, do you need to log the time for a) certification/currency purposes, and/or b) as training received? And if you do log it, is it "dual"?
For certification/currency, it's usually best to log everything: even if you don't need to show the FAA something, your insurance company or FBO might ask you for details of your flying time.
As for training, in order for a flight to be "training received", the instructor has to endorse your logbook:
(2) The training time must be logged in a logbook and must: (i) Be
endorsed in a legible manner by the authorized instructor; and (ii)
Include a description of the training given, the length of the
training lesson, and the authorized instructor's signature,
certificate number, and certificate expiration date.
That's a simpler case: if the instructor didn't sign your logbook then you didn't receive training.
But let's say that you decide to log the flight for one or both of the reasons above. Is it "dual"? Well, since there's no official definition of "dual" then it depends what you use it to mean.
Again, practically speaking, most people use "dual" to mean "training": my logging software uses it that way, and many paper logbooks seem to as well. So I only log dual if I have a corresponding endorsement from my instructor. (The software also has options like "instructor on board" or "introductory flight" that I can use for the scenarios you described.)
As a specific example, I did an FBO checkout ride in a new aircraft type a few months ago. I was already rated for the aircraft category/class and the instructor with me didn't endorse my logbook, he just updated the FBO's internal documentation to state that I'm now allowed to rent it. So I logged the time as PIC, but not dual.