How can I define the three axes and their purpose in an isometric projection?
Let's imagine we are going to paint this bowl of fruits on the table that contains an apple, a banana and a pineapple. To help us with the borders, we put this wooden frame around the bowl. As you can see the whole bowl with all the nice fruit can be seen within the frame. If you close one of your eyes, you will even be able to see the picture that we are supposed to paint.

Now, have both of your eyes open again and you see the real bowl of fruit within the frame.

Get up close on the frame and see that the banana is actually extending out of the frame and towards us.

These are the axis of projection. The X and Y are parts of the frame. X is the long bottom part. Y is the left/right shorter parts, the verticals, going up from the bottom to the top.

The third axis is how the banana is protruding against us. We could put a ruler in the left lowermost corner of the frame corners just to be able to measure how far the banana or the pineapple leaves are protruding at different places. That ruler is Z.