I have recently watched two YouTube videos on this subject, that you can see here (first by "Science Asylum" and the other by "Veritasium").
So, both videos ask the viewer to imagine a simple circuit, a battery connected to a resistor (such as a light bulb). As the battery is connected in the circuit, it will cause the electrons in the wire to drift away from the negative terminal of the battery, dissipate energy in the light bulb and then be attracted into the positive terminal.
Veritasium video shows though that one shouldn't really think of this as some kind of "domino" effect for electrons, but rather as an EM wave propagating directly from the battery to the light bulb, as computing the Poynting vector reveals that the flow of energy is directed away from the battery and into the load (as shown in this screenshot taken from the video)
This is used in the video to show that even if the wire segments are, say, one light year long, if the battery is 1m away from the light bulb it will take only 1/c seconds for the light bulb to turn on (watch the video if this sounds confusing, he does a great job explaining this).
However, in the video by Science Asylum there seems to be another perspective on the subject that I don't see exactly how it is equivalent to Veritasium's. Science Asylum shows that the electric field on the wire is not radial to the wire (as it seems is the case in Veritasium video, which I don't fully understand), but rather parallel to the wire (which kind of makes sense, as Ohm's law says that J=\sigma*E where \sigma is the electric conductivity, which means that the current density is in the same direction as the electric field...). But this means that when one computes the Poynting vector (as Science Asylum does in his video), it points AWAY from the wire and into the "field" around it. He then basically says (from my understanding) that the energy flows from the battery into the field around it and is then picked up by the light bulb and the wires directly from the field (doesn't say much about how "fast" is this process as in Veritasium video...).
In any case, what is the correct picture? I don't understand how both can be correct, given that they show electric fields pointing in different directions in the two videos... Is there something I'm missing here?