The quick answer to your question is yes, you need 240v service to get 240v. So you either need to have a 120/240v delta transformer for incoming power from the main line, or you could use a step-up transformer, or you could have a 240v Y transformer (I just don't remember what your phase to phase voltage is in that case).
The measurement on Van's graph that indicates 240v is a peak to peak measurement which doesn't exist in reality. It doesn't measure and usable voltage as you don't really have voltage between a phase and itself.
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Given that, you probably don't want to connect a device designed for 220/240v to a 208v single phase as you will definitely loose performance, and you really don't want to connect it to split-phase as it may not be designed to handle that. ...
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