What specific markers does a Covid-19 PCR test look for?

The SARS-CoV-2 diagnostic tests that are currently in use, both serological and nasal swab, are binary tests - they either are positive or negative, like a pregnancy test. They do not (yet) report viral titre, like the first image you linked, or look at restriction enzyme "gene fingerprinting" patterns, like the second. They simply tell you if either a part of the virus is in the sample (RNA or capsid protein in the two nasal swab tests) or if your immune system has seen enough of the virus to mount an antibody-mediated immune response (the IgM/IgG blood tests).

It's worth pointing out that all of the current tests in the US are available under emergency use authorizations from the FDA, meaning they haven't gone through the usual (and time-consuming) process of full qualification and validation. That type of work needs to be done in order to develop quantitative testing ("You have a viral titre of X, compared to an average range of Y through Z."), which will likely appear in the coming months and years as the proper development work is done to support their validity.

To my knowledge, all of the PCR-based tests that have been authorized for diagnostics use fluorescently labeled reporter probes specific to the the SARS-COV-2 viral genome. In real-time PCR, the intensity of this fluorescent signal is measured (spoiler alert) in real time, after each amplification cycle. The fluorescent signal increases with the amount of the target sequence present in a reaction. If the signal exceeds a certain baseline threshold before a certain number of PCR cycles, the test is considered positive for that target. It really is as simple as that, but a lot of work goes in on the development and validation of these threshold cutoffs to ensure they are sufficient for sensitive detection.

What identifies the readout as unique to this virus is the design of the probes and the primer sets that go with them. These are designed to be highly specific to a given target, and are validated in silico against existing sequence databases, and also empirically against closely related organisms, or just against other things that might be present in a specimen. They should also target conserved regions of the genome that are unlikely to mutate. Most of the tests I have seen use at least two different SARS-CoV-2-specific probe targets, and at least one amplification control (often targeting a conserved human gene) to ensure that the reaction enzymes are functioning properly. But the specific regions of the genome that are targeted can vary between different tests.

Using real-time PCR saves clinical lab techs the time and effort of making gels and running the amplified PCR products out on the gel, like those in the second example you posted. This not only improves the turnaround time for testing results, but saves countless work hours for the clinical labs processing these tests, especially when you consider the scale of how many tests are being processed right now.

As with any nucleic acid test, a positive result can only indicate if the target is (or was) present. It does not provide information to clinicians about whether it is actively replicating or even infectious. But in general, this kind of testing is highly sensitive and specific when properly designed and validated.