The more DIY methods might not require a bigger motor, however they're not recommended unless you want to be constantly having to adjust things to try and get consistent and accurate performance.
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Different size motors on each axis are not a problem. Whatever controller you end up using, is only interested in the number of steps it takes to move the axis a set distance. The controller is only responsible for sending the correct number of steps to the motor drivers, and has no interest in what the motor driver is actually powering.