"Payroll tax" is often used to refer to the Social Security tax, when one wishes to obscure the nature of the tax (ie that it funds the Social Security program). "Withholding tax" is more normally used to refer to federal income tax based on the income tax tables.
What the Senate bill did on the Social Security tax was set a limit of $18,350 (1/6 of 110,100, the limit for the entire year) for the first two months of the year that the 4.2% rate applies to. 6.2% applies above that limit. After the first two months of the year, rates are yet to be determined.
So instead of a simple calculation involving a single rate and a single wage limit for the employee portion for the entire year, we have three rates, two for the first two months and one for the last 10 months, and two limits that apply. For reporting the quarter totals (Form 941) employers would need to report the total wages for the first two months below $18,350, and the total wages above $18,350 along with the total wages for the third month in the quarter.
I would be very surprised if any payroll software were capable of handling this calculation for the Social Security tax right now. Up until now, there was no reason to build such a ridiculous calculation into your payroll software.
It seems like to me that it used to be that politicians, or at least their staff, had some inkling of the real world effect of changing tax policy, and the need for lead time for the IRS and SSA not to mention software vendors to adjust to these changes. At the worst, changing tax rates, calculations or reporting requirements should be done at the end of quarter if not the year.
News reports suggest to me that the only sticking point between the Senate and the House is how to fund the bill. The two month time frame apparently came about because that the Senate and House only agreed to funding sources that totaled enough to continue the tax break for that period of time.