Some confusion here. Split-phase (230-0-230) does have 460V between lines. That is an old system, more or less obsolete in the UK, where larger LV installations were served from single-phase HV lines such as on farms. It was not used where 3-phase supplies were available. Two phases out of a 3-phase supply will have 400V between lines and this will be the case in an urban system where 3-phase distribution cables are laid in the road.
Re the installation mentioned in the OP, if the service cable is old (e.g. prewar) it was likely a 3-core cable installed for 3-wire DC mains. A small installation would have only one outer (line) and the neutral, hence a 2-core service cable, but a larger one would be supplied with both outers via a 3-core service cable (i.e. positive line, negative line and neutral). When changed over to 3-phase AC, the two lines of the existing serivce cables were connected to two of the three phases of the new distribution cables. I have this at my house, with two separate single phase supplies derived from the two lines. I cannot have the third phase without the 1930s service cable being replaced.
FWIW there was another historical system properly called '2-phase' that was neither split-phase (which is really just centre-tapped single-phase) nor two phases out of three. It used a 3 or 4 wire supply with the two lines at 90°, intended to give the advantage of a rotating field without power-zeroes in induction motors, but it lost out to the more versatile 3-phase system.