Hi brookesy,
I knew Flecc would beat me to it

but here's what I've put together, it may repeat some of what Flecc said, in a way, and I can't vouch my additions are 100% correct

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I'm definitely no expert, but I'd say the main factors are the power-to-weight ratio of the bike, and the way the "controller" manages the motor power - especially from a standstill, which can require a lot of power and so on some bikes power is introduced only once the bike is moving, as a means to conserve some energy.
But what's the power level of a bike? the 200W "continuous power" rating (the power level at which the motor can happily work continuously) of many motors doesn't give the information needed to quantify a bike's "peak power" output, which, depending how its managed by the controller, determines both it's acceleration and hill-climbing ability, and depending on the motor can be a little more than 200W or a lot more (e.g. ~300W-700W) so that's one reason why different "200W" rated bikes can behave very differently in that respect.
I think yet another factor is the type of motor - older motors are "brushed" and apparently can give more torque than newer "brushless" motors in some situations, so that may also be why many newer bikes aren't "cutting the mustard" compared to your Sakura
You also said before that you thought your Sakura's previous owner may have modified the bike for higher power output, so if you had more information on that it may give you a better opportunity of comparing apples with apples

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I'm not sure if the Sakura uses a brushed motor, or if its similar to the Powabykes, but they apparently have a very powerful motor - about 700W peak - which would help account for the acceleration it gives.
In case you haven't seen it and it's useful, have a look at
this video from Italy (posted by Leonardo the other day) showing a first look there at the ezee Torq, Liv & Forza: its by no means a fair comparison of the 3 bikes, the Torq in particular sounding "strangled" at low speed! but it may give you some idea of how these bikes are at low speed or standing starts.
To be fair, I think the demonstrator goes easier on the throttle from standstill for the Liv & Torq than the Forza, so less acceleration is seen, but you can clearly see the battery meter drop down into the red on the Liv due to the power drain from standing start, and the Forza's good (I think) acceleration from standstill.
You may also be able to notice the different character of the Liv's brushed motor compared to the Forza's (& Torq's) brushless motor.
Hope that all helps & is not information overload brookesy
Stuart.