This is a very good conversation.
I do not consider myself as any kind of expert.
However, I thought that I should share a trick that I discovered.

If A1 is the cell with the date, then I have used the following formula in the conditional formatting:
=and(DOW(A1)=7, mod(CEILING(day(A1)/7),2)=1)

It worked for me. This highlights the second and fourth Saturdays.

Girish Mahajan