Antennas always have two wires going to them. A single wire going only to a 1/4 wavelength whip is not an antenna.
Mike, you’ve got the right idea, but here are some interesting details that you might not be aware of. The point where RF power is fed into an antenna system is called the feedpoint. The amount of power that a transmitter can push past that point depends entirely on the impedance of the transmitter output (the source impedance) and the impedance looking into the antenna feedpoint. Remember your Thevinin equivalent circuit. If the generator (the transmitter circuit board) were infinitesimally small, and the antenna were any length of wire, the input impedance would be near infinity and no power would transfer from the transmitter to the antenna. This is the point you are making. However, as soon as we provide any conductor of sufficient size on the transmitter side of the feedpoint, the antenna will couple to that as its “counterpoise” automatically and the feedpoint impedance of the antenna will automatically come down from infinity to some lower value. So the transmitter board automatically becomes part of the antenna system, whether we like it or not.
Now, if the transmitter board has a ground plane that is large, the antenna system feedpoint impedance comes down to a very useful and practical number. For example, with a very large ground plane that is perpendicular to the antenna wire, and when the wire is nearly ¼ wavelength long, we get a practical impedance somewhere near 50 ohms. Transmitter boards are usually not very big, so we usually have a compromised ground plane, but even in these cases, the feedpoint impedance drops to a useful value, often below 120 ohms. In a case like that, the antenna will work pretty good, just not optimally. And yet, it seems to have only one wire. The other wire is actually the ground plane of the pcb.
Now, here’s another interesting example. You remember early cellphones that had the pull-out whip antennas? These antennas were often simply ¼ wave whip antennas. These seem to have only one wire so how do they work? The other wire is the rest of the phone. The main circuit board acts as the ground plane for the whip antenna, that’s how it works.
So, whenever you see a transmitter board that has one wire connection called ANT, you can assume that the “second wire” is the board itself. And we hope the board is large enough to do its job as part of the antenna and when it isn’t, well, we just live with it.
One other point worth mentioning is about your statement “a single wire going only to a ¼ wave whip is not an antenna”. Indeed, the wire going to the whip becomes part of the whip. In the case of this board, the traces and pins that carry ANT signal from the module, through his board, to his whip are also part of the antenna and their length should be calculated into the total length.