To answer your question, as F.4.1 stipulates:
"A = area resisting the compressive force, as illustrated in Figure F-2", or if you prefer, it is the "compression ring area". It is a geometrical calculated area- but the boundary of area came from other considerations not explained there.
Anyway you have to consider Figure F-2.

I have to recognize that the Pf calculation procedure described in API 650 is not easy to be understood.

In fact, failure of the roof-to-shell junction can be expected to occur when the stress in the compression ring area reaches the yield point, as F.6, Calculated Failure Pressure says. That's why with the notations of Addendum3- 2011, the failure pressure would be expressed in SI units as:

Pf=8*Fy*A*tanθ/D^2+4/PI*DLR/D^2 - units in [Pa]
(where A is the area subject to stress Fy)

or

Pf=Fy*A*tanθ/(125*D^2)+0.0012732*DLR/D^2 in [kPa]

However, this is not so simple explained in API 650.

As you remarked, API 650/F6 prefers to to express "Pf" in terms of "P".

Just to explain the connection between P and Pf (and hoping I'm not inducing more confusion!), F.4.1 considers a safety coefficient of 1.6 applied to the pressure effects term (i.e. the first term), so the maximum design pressure, P is written as:

P=Fy*A*tanθ/(1.6*125*D^2)+0.0012732*DLR/D^2=
=Fy*A*tanθ/(200*D^2)+0.00127*DLR/D^2

and this is the maximum design pressure, P, for a tank that has been constructed or that has had its design details established; the basis of that derivation was to include a coefficient of safety of 1.6 addressed to the pressure term in the "failure" pressure expressions.

Instead to calculate directly, API 650/ F6 prefers to express indirectly Pf in terms of "P", so would be:

Pf=Fy*A*tanθ/(125*D^2)+0.0012732*DLR/D^2=
=1.6*P-0.6*0.0012732*DLR/D^2=
=1.6*P-0.0007639*DLR/D^2
(you can see that they mistyped 0.000746 instead of 0.000764!)

You can conclude that API 650 prefers to calculate first P and later Pf instead to calculate directly Pf.

You may note also that, in the previous editions of API 650, instead of DLR appears a term based on the thickness of roof plates and DLR is calculated with a metal density of 8000 kg/m^3 (8 times the water density).

Best regards.