All that matters to you is does the starter work when the button is pushed. If so then leave it exactly the way it is.
The starter solenoid circuit can be double ended in the way it gets wired. Some have power going in the yellow/red and others have that wire as the ground. It can work either way as long as the other wire needed goes where it needs to as well.
Whether something looks OEM is meaningless if you understand how the circuit works, I have reversed the wires more than once to save $$$$ to use a $20 solenoid that works fine instead of an overpriced one at like $80. Just to look 'OEM'. There are internal grounded solenoids as well as external, either will work as long as wired correctly. I've been using a $20 riding lawn mower solenoid on mine for many years. Works perfectly.
Honda changes that circuit up too, the SOHC is backwards from the DOHC in that exact yellow/red wire useage. The SOHC uses it as the ground for solenoid, the DOHC uses it as power to the solenoid. It gets flipped on the later models to make the grounds partly green (green/red), it lends to recognizing green as a Honda ground color and the extra starter interface switches complicate the ground wire more anyway. Using yellow/red there would confuse things more.
I would suspect the handlebar starter button to be wired slightly different based on that solenoid mod.