Will it simplify things if I point out that the bottom note of a Time Signature tells you how the 'beats' are written.

When the bottom number is 4 the beats are written as quarter (1/4 get it?) notes or quavers.

When the bottom number is 2 the beats are written as half (1/2 get it?) or minims.

The top number tells you how many beats are in each bar.

It's the beats that differ not the way the notes are written. If you have a notation processor like Sibelius try pasting a bar of 2/2 into a score of 4/4. They will look the same. They wont sound the same unless you make sure the BPM for 2/2 is half that of 4/4.