Let's review what is really happening: "My issue is getting the payroll liability to show up weekly in the Payment History under payroll liabilities screen."
It Shows because of each Pay Date. It accumulates based on the Payment Schedule you told the system you need.
"In the payroll setup screen both the employee and the employer match are set up to come out weekly on Friday for the previous week's Liabilities (Monday thru Sunday)"
It's not the Previous Week's Liabilities. The Liability Clock starts ticking because of the Pay Date. Let's review the use of the terms and calendar for Payroll:
Pay Period is Mon - Sun
Pay Date is every Fri <== That is the Liability Date, because Money Happened and the means taxes are "on the clock now"
Then, the Payroll Item is Scheduled for when and how often it needs to be paid. Example: You pay Once a Month. That means ALL paychecks in that one money are accumulated into One Entry for Pay Liabilities, having the PERIOD of Jan, with a Due Date in Feb, for FICA typical example.
You don't see Weekly Deductions in Pay Liabilities, because you set it to be paid Monthly, for a Month's accumulated amount.
"In the payroll item listing - again both of the Employee and Employer are set up, but I think the area that is causing the issue is the Default Rate and Limit. We get paid every week. the Limit Type is only for Annual-Restart each year, Monthly-Restart each month or One Time Limit."
You are Confusing different things that need to track against dates or calendar functions. You deduct from me based on my participation; you match based you your participation. The Calendar Limits are Federal Statute. The Weekly Paycheck is the Financial Event. The Due Date and Cycle of the payment is the Liability Check requirement for you to Comply with or be considered late.
"I think the reason it worked the first time I did it was because, I chose Monthly and it was the start of the month."
Not really. You are confusing too many Perspectives that all happen to be Date-driven. That is Payroll. And the function for Pay Liabilities does not relate to the Paycheck Cycle, but what you Told the system you need its help Managing, to keep you on track.
Let's do one Extreme example:
The Employer knows that FUTA is .6% on the wage limit of $7k per employee; that means in the entire Calendar Year, you will never owe more than $42 per person. The Payment Rule is to pay when it reaches $500 total, or by the end of the year with the 940 form you file. You set the system for "Schedule this liability as Annual, because there are only 4 of us, and even if my staff completely turns over this year, that will still be less than $500 by year end."
So, Payroll is Weekly.
FUTA is Annually.
That means in the Payroll Center, you see one line for FUTA set for the YEAR as the PERIOD, for 2019. It Increases every Friday, when paychecks are Saved. There are not Weekly Lines for FUTA, in Payroll Center. There are Weekly Lines in the Payroll Liability Detail report and that account, which explains that Accumulated amount you see in the Helper tool of Pay Liabilities, which increases every time you issue a paycheck.