Can a television detector van REALLY tell if you have the TV switched on?

No, they were not lying to you.

Televisions depend on electronic circuits which resonate with the signal coming in so as to decode it.

TV receivers are also small antennas. The incoming signal resonates on the circuits and this creates a secondary pulse, which will re-radiate in a different direction from the incoming signal, just because of the nature of circuits and an antenna. When a TV is tuned to a carrier frequency it also re-radiates it more or less in all directions from its receiving antenna. From this, sensitive detectors can detect the location of the TV's receptor antenna if it is active.

Agree with Anna. The part of the signal that is radiated outwards is the local oscillator (LO) signal. That determines what freq you are tuning to. There are isolators and other components (e.g., one way amplifiers, switches, filters, etc) that will bring that signal down before it gets to the antenna, but not all of it. That LO is used in the mixer to down convert the incoming signal, so it can also follow that same path out (to some extent). If the LO is heavily filtered, a harmonic or cross term can be used.

TVs don't have to be highly isolated, and the LO signal is such a narrow bandwidth you can use narrowband filters to reduce the noise enough and get a high enough SNR. And by the way, those vans or whatever can have directional antennas to additionally increase the SNR.

I understand that that used to be the main way to do TV polls, but nowadays enough people agree to have a device that reports it.

This is explained in detail here

In other words, the resistance of the load circuit must match the radiation resistance of the antenna. For this optimum case, $$P_{\text{load}} = P_{\text{rad}} = \frac{V_0^2}{8 R_{\text{rad}}}=\frac{P_{\text{in}}}{2}.\tag{1108}$$ So, in the optimum case half of the power absorbed by the antenna is immediately re-radiated. Clearly, an antenna which is receiving electromagnetic radiation is also emitting it. This is how the BBC catch people who do not pay their television license fee in England. They have vans which can detect the radiation emitted by a TV aerial whilst it is in use (they can even tell which channel you are watching!).