Confusion about FCF deriving from Cash Flow from Operation and Accouting profit

Dear All,

In deriving Free Cash To Firm or Free Cash Flow to Equities. There are essentially two approaches

1. Approach one: Using EBIT or PAT as the starting point.

FCFF: EBIT(1-T) + Non-cash expense – Investment in working capital – Investment in capital expenditure.

FCFE: PAT + Non-cash expense – Investment in working capital – Investment in capital expenditure + Net borrowing

2. Approach two: Use Cash Flow from Operation (CFO) as the starting point.

Assuming interest paid and tax paid are deducted from CFO.

FCFF: CFO + Interest paid – Tax saving on interest paid – Investment in capital expenditure

FCFE: CFO + Net borrowing

My questions is if Free Cash Flow is in real sense a sort of cash flow.How come USING accounting profit adjusted by interest expense and tax saving on interest expense correctly reflects the impact of INTEREST and Tax paid on cash flow.

Are we better to use the ACTUAL interest paid and tax paid as WE do in approach 2?

For me, Obviously, USING CFO as the starting point will be different from using EBIT or PAT in their treatment to tax and interest.

However, in most texts, the fore mentioned two approaches are considered being equal

Is the government tax rule generally assumed to be the same as accounting tax rule when we deal with free cash flow and hence there is no difference between interest expense (tax expense) and interest paid (tax paid)?