When we run payroll and want to reinvest a part of our income back into the company, is making a journal entry to move the deduction from payroll liabiltiy, which is currently the only way to withhold the funds from the paycheck, to the Equity Contribution account the only way?
No, any deduction for any purpose can be withheld from net pay upon agreement of the employee.
Please verify that you are in fact a S Corp or LLC electing to be taxed as a corporation in order for ay owner to be on payroll. And since you effectively are not a member but a shareholder your equity contribution might have to labeled as purchase of shares.
We know that what we are doing is correct. I'm looking for the machanics behind what I'm doing within QBO Payroll. I appreciate you letting me know that there isn't another way. Maybe that's something they can add for the future.
You can set up voluntary payroll deductions as per the attached https://community.intuit.com/articles/1770957-set-up-voluntary-deductions-in-online-payroll and assign the value as zero, but the destination account as your owner equity contribution. Then each payroll adjust the voluntary (after tax) deduction to equal net pay - resulting in a zero dollar paycheck.