Yes, I 'd say you pretty much have it. More specifically, suppose I have a regression model with one independent variable:

Yhat = b0 + b1X1.

I can determine the R-Square (R1^2) and F (associated with b1) as

F = (R1^2 / k) / ((1 - R1^2) / (N - k -1))

where k is the number of independent variables.

Suppose I add a second independent variable (X2):

Yhat = b0 + b1X1 + b2X2.

We can determine the change in R-Square (deltaR^2 = R12^2 - R1^2) and change in F as:

F = (deltaR^2 / (2 - 1)) / ((1 - R12^2) / (N - 2 - 1))

where the change in F tests the unique contribution of X2 (i.e. as if X2 was entered last into the model) and R12 is the R-Square associated with the second model.

Note that: + - Sqrt[ deltaR^2 ] is the semi-partial correlation between Y and X2.