Simple COUNTIF with OR
Here is what I am trying to set up:
Count the no. of cells in Column A where condition X or condition Y is true. How do I write this in a formula?
Answers

More info is likely needed, but it would be something like this:
=COUNTIFS([Column A]:[Column A], OR(@cell = "Condition x", @cell = "Condition Y"))

What would @cell be?
If this is my data set, I want the "Apples" sheet summary field to count where "Apple" or "Apple, Orange" is true
Similarly, I want I want the "Orange" sheet summary field to count where "Orange " or "Apple, Orange" is true

@cell & @row are useful functions in Smartsheet to create more efficient formulas. More info here: Create efficient formulas with @cell and @row  Smartsheet Learning Center.
I would use a formula like this for the above scenario:
=COUNTIFS([Column A]:[Column A], OR(CONTAINS("Apple", @cell), CONTAINS("Orange", @cell)))
After I thought on it more, you probably need it like this:
Apples: =COUNTIFS([Column A]:[Column A], FIND("Apple", @cell) >= 1)
Oranges: =COUNTIFS([Column A]:[Column A], FIND("Orange", @cell) >= 1)
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