I am still confused about the meaning of "phase" in the context of typical North American household wiring. Is anyone able to explain the difference to me?

My understanding is that a typical household is wired with a "Single-phase, 3-wire, common neutral" power supply from the utility company. The power coming from the high voltage line at the power pole is a single phase from the distribution grid. The transformer serving each household steps the power down to 240 volts, which is split into 2 wires carrying 120 volts each, and a center-tap neutral off the transformer. These are the three wires coming into our main distribution panel. Thus the red and the black hot wires are actually different poles of the same phase.

Electrical discussions sometimes say you can share a common neutral between different phases, since the neutral wire sees the load at different times in the cycle (120 degrees out of phase). My understanding is that this is only true in a 2-phase power supply. However, hooking up a common neutral to circuits fed by a black and red wire in a typical household would be dangerous since in fact the neutral is now carrying 240 volts from the same phase.

Is my understanding of this correct, or am I confused somewhere? I keep reading stuff, and the terminology does not always seem to be clear.....