If you have two hots and a neutral of a three phase system, then you do have two separate phases available.
A load connected between any two conductors of a three phase system sees a single phase, but as soon as you have three conductors (any three, including the possibility of two hots and the neutral) then you have different available circuits with different phase angles available.
You are correct, though, that this is different from the 'two phase' systems that were used in the past. In a 'two phase' system you have two separate circuits where the phasing difference is 90 degrees rather than the 120 degrees found in 3 phase systems.
Thread hijack going to new odd systems: EPRI has done research on using 12 and 18 phase systems for power transmission. For a given phase to neutral voltage, the phase to phase voltage goes down as the number of phases goes up. (I can give you the math, but imagine a circle. The radius of the circle is proportional to the phase to neutral voltage. Now draw a set of equally spaced dots around the circle, one for each phase. The distance between the dots represents the phase to phase voltage.) Because the phase to phase voltage goes down, the required phase to phase spacing goes down. The spacing from the conductors to the tower and ground remains the same, but you can pack more conductors into the same right of way, and thus carry more total power.