You are correct that the words "Class II" and "Class III" are not included in the NFPA fire alarm power limited requirements. The relays on the circuit board in question, however, are designed by the manufacturer to switch power limited circuits. Yes, many of the circuits that are switched by the relays are Power Limited Fire Alarm (PLFA) circuits, but also included in the possible switching are Power Limited (PL) circuits that fall under the non-fire alarm power limited circuit requirements.
In laymen's terms, the word "Listed" means "The device, circuit, process, or instructions has been tested for use and found safe by a third party, nationally known testing laboratory like UL, ULC, CE, CCC, FM, etc., and it has been placed on the laboratory's list of safe items." Listed, in other words, includes anything (including installation instructions) that has been placed on the laboratory's list - it has been tested and found to be safe for use. Also, if anything is on the list, the testing laboratory's stamp is shown on device, circuit, process, or instructions.
To be used for fire alarm systems, all equipment, including the installation instruction sheets that come with the equipment, has to be "Listed" for use. On the Listed installation sheet that comes with the relay board for the fire alarm control panel are the words "MUST BE CONNECTED TO A LISTED POWER LIMITED SOURCE OF SUPPLY". Because the words are "POWER LIMITED" and the words are not "POWER LIMITED FIRE ALARM", the requirements for the power supply fall under the Power Limited rules rather than the Power Limited Fire Alarm rules.
Low Voltage 24-Volt Transformer versus Class II or Class III 24-Volt Power Supply
The question about the power source revolved around the idea that a "Low Voltage" 24-volt transformer would meet the "LISTED POWER LIMITED SOURCE OF SUPPLY" requirements shown in the NFPA. The problem is that just-any 24-volt transformer doesn't legally meet the code, but even worse, by itself, just-any 24-volt transformer isn't safe as a power limited power supply (that's why it doesn't meet the legal code).
Many times, and from many people in all levels of the power limited community (not just the fire alarm community), I have heard the words "low voltage" used to mean "power limited". I know this is not correct, you know this is not correct, but a very large number of installers, technicians, and even designers think that "low voltage" is the same as "power limited".
You can burn down buildings using the wrong low voltage 24-volt transformer. However, a Class II or Class III 24-volt transformer won't burn anything down. Low voltage does not equal power limited or fire alarm power limited.
When I was asked the question of whether a low voltage 24-volt transformer could be used for a "LISTED POWER LIMITED SOURCE OF SUPPLY", my answer was to instead use a transformer with the words "Class II" or "Class III" stamped onto the case of the 24-volt transformer.
Using a Class II or Class III transformer provides the confidence that the transformer is safe, and that it does meet the NFPA 70 Power Limited Code.