Notice on the right. I chose C major
In C major, iiim is often replaced by III, which means Em becomes E major
And Am turned into A major.
But the right side itself does not include E major and A major.
Then on my left, I chose A minor, which resulted in E major and A major, but the sound was different.
Is there a better way to solve it? I actually want to choose a C major and display the commonly used chords.
I’m not sure I understand the question in relation to this diagram.
On the right hand side, the vertical columns show the harmonised diatonic scale, and successive rows show the parallel modes of the root, which is C.
The harmonised ionian mode (top row) always shows a sequence of major / minor / minor / major / major / minor / diminished whatever the key, and so in this case, D and E will be shown as minor triads. Am is shown as the 6th from that same sequence.
On the left screen cut, A major is shown as the root triad of an A ionian mode, E major is shown as the 5th of that mode as you would expect. The Amin top left is shown as the root triad of the aeolian mode.
Whereas modes set out using the ‘derivative’ approach ** all have the same notes, those using a parallel approach as here just use a common root, so have different notes.
So the diagram on the right does not show A major or E major because those chords don’t appear in the harmonisation of the scale, so Scaler won’t show them in this display.
Since E major and A major in a C major progression would be ‘borrowed’, you are better looking at the Secondary Scale’ on the Mod page, which will give you path ways to bring them in. So in C ionian, you might go F maj / F# min / B7 / E major. A major is then the 4th in the harmonisation of E ionian, where you now would be.
It can be quite useful to look at the (subtle?) diagonal lines on the modal interchange diagram you show; if you look at the top right corner in the right hand side of your screen shot, you will see that it is B dim. You then see if you move to the left one and then down one, you have A dim. Do that again and you have G dim, all the way down to C dim bottom left.
To me, the pattern is pretty clear when you look at it from that perspective, and that’s because the scale intervals ‘rotate’ to the left through the modes. So where the major scale is TTSTTTS, the next mode (Dorian) is TSTTTST. So as in the major scale top row the B dim triad was the last ‘S’, you can see that in Dorian (second row) the S has moved one to the left. With this ‘parallel’ presentation the notes are always in the same vertical position, e.g. A min in C major becomes A dim in Dorian. Looking at it this way for me means that you don’t have to think about it much to see where the notes have gone and what the triads are.
**
Whereas the parallel presentation of modes will show (as here) C maj, C min, C min … etc, the ‘derivative’ approach would show C major, D Dorian, E Phrygian, F Lydian,G Mixolydian, A Aeolian and B Locrian. So now all the dim triads would be in a vertical column, but the note headings for the column would rotate.
Not sure of any of this helps or not … 
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