I went to Apple Authorized Repair center. They DID manage to run Apple Diagnostics Tests (unlike how others and myself have reported this is not working from Cmd-D) - it showed no errors from top level view.
To clarify: Apple has two different "diagnostic" utilities: (1) Apple Diagnostics (note plural) / Apple Hardware Test, which is available to users; (2) Apple Service Diagnostic (singular), which is used in-house by Apple and its Authorized Service Providers (AASPs).
1. Apple Hardware Test was provided in various forms (on System installer discs, installed with the original OS, or accessed via the Internet) with Macs from <2000 until June 2013 (thus that's what runs my Early 2013 MacBook Pro – used to, anyway). After June 2013 (e.g. the Late 2013 MacBook Pro), Apple Diagnostics is either accessed via the Internet or from the Recovery partition in High Sierra or Mojave (as I discovered in my research).
AHT/AD is accessed by starting with the D key depressed (not ⌘D, which at least in Mojave will default to starting in the OS). AHT has a primitive-looking interface and offers either a short or an extended test. AD starts automatically and shows a progress bar with the legend:
Checking your Mac… 3 minutes remaining, 2 minutes remaining, About 1 minute remaining.
These utilities will find gross problems, but don't go into the hardware in any detail. Neither can be loaded/run in Macs with the Cannot load 'EFI/Drivers/TestSupport.efi' error discussed here.
2. Apple Service Diagnostic is Apple's in-house diagnostic software. It was in use up to mid-2014; as the article says, its various versions (for different Mac models) can be obtained here and there from the 'Net. Since mid-2014, the current version (name unknown) must be accessed via the Internet from Apple (i.e. is not downloadable), and requires a password, so is accessible only to Apple shops and AASPs.
ASD includes both an OS version (running in a stripped-down MacOS) and an EFI version (running like AHT/AD from a basic EFI boot that employs some firmware built into Intel CPUs). Each includes a long list of tests, and takes from 30 minutes to an hour or more to run. (The major advantage of the EFI version is that it can test more of the memory, as practically no memory is occupied with running the software, whereas the OS version occupies a substantial chunk. The OS version otoh allows running user-interactive tests.) The EFI version will not load/run in Macs with the Cannot load 'EFI/Drivers/TestSupport.efi' error discussed here, but the OS version will, since it uses an OS, not just the EFI drivers.
I'm guessing that the current Apple in-house diagnostic software (which I've never seen) also includes both OS and EFI versions. From the photo you've posted, my guess is that the AASP (?) you took your iMac to was able to run the OS version (but maybe not the EFI version?), which clearly does a comprehensive testing of the system. Also it appears that the current software tests older Macs (such as your 2012/2013 iMac) that predate its introduction in 2014.
BTW the yellow warning on memory is because of the 3rd party Kingston HyperX 16GB I just swapped in - non Apple certified.
And I'll bet it cost half of the price of the Apple equivalent? Kingston is not exactly a fly-by-night operation.
They also tried to use a ROM update tool…
Well, the ROM update may have something to do with the Cannot load 'EFI/Drivers/TestSupport.efi' error; still not clear. Why the ROM of the 2014 MacBook Pro was not updated by installing Mojave (my report above) is a mystery. But the Cannot load 'EFI/Drivers/TestSupport.efi' error did not appear either, so thusfar nothing is either proven nor disproven.
Annoying that theirs works and ours (or even the network based consumer ADT doesn't!)
Yeah, ain't it?