Fine line really, I don't think you're wrong per se, remember too turbos are considered 'forced induction', more like we see it from different angles.
Yes there will be an element of push when the turbo is on boost. It's a bit chicken and egg, the turbo won't be on boost unless the engine is accelerating, think of an engine on tick over, the turbo is spinning, but not boosting/pushing/compressing to any great degree. The engine is still running, ergo the sucking action is from the piston strokes as already described. Things change when accelerating as the sequence of events drives up the boost pressure, inlet charge compression etc etc. Please also remember it's a combination of factors including the exhaust stroke causing an element of vacuum in the cylinder, which complements the action of the falling piston sucking in the inlet charge.
There's so many other elements to think about, the turbo does make a difference because the piston is sucking in air which because it's compressed is also more dense which causes better burn, it's probably cooler too which again lifts the burning capability.
I can't verbalise the way that site does, so for a full explanation much better than I can, have a look at the how it works stuff.