Ooops!
I gave you the wrong the formula!
Let's sort this out...
If you have 2 formats applied, green and red, what color is the cell to
begin with as the default?
Normally, white is the default color. If you want to set fill color with
conditional formatting you start with the a default color then as the
conditions are met the cell color changes according to those conditions.
I have a hunch that maybe your RED condition isn't set properly. Try it like
this:
=AND(COUNT(A1),A1<TODAY())
An empty cell evaluates as 0. So, if you simply test an empty cell to be
<today that condition will be true and the format will be applied. You need
to test for 2 conditions. 1) the cell does in fact contain a date (number),
and 2) the date (number) in the cell is <today. The above formula checks for
both of those conditions.