That's the technical details. You only need to concern yourself with how you use it.
For greyscale, both the buffer and back-buffer contain data used for drawing to the screen.
For 3-level:
00 - white
01 - grey
1x - black
For 4-level:
00 - white
01 - light grey
10 - dark grey
11 - black
where the first bit is what's present in the buffer and the second for what's in the back-buffer.
Cross-hatching and evolving backgrounds refer to the techniques to reduce flicker.

yeah, where each sprite contains part of the data, as shown above.

Manual interlacing sounds awful and editing them sounds even worse...
Thankfully, Axe does this automatically.

So, now Im having another issue.

When I try 4-level grey I see a checkerboard-ish pattern. It happens on my nspire and my sis's 84+.

Here's the code:
.GREY
DiagnosticOff
SetUpEditor
ClrDraw
ClrDrawr
[FFFFFF000000FFFF]->Pic1
[FFFFFFFFFFFF0000]->Pic2
0->G
For(G,1,500)
Pt-Off(0,0,Pic2)r
Pt-Off(0,0,Pic1)
dispGraphrr
end

I'm expecting:
Black Bar
Light Grey Bar
Dark Grey Bar

and I get them, but there is a distinct checkerboarding that doesn't occur when I do 3-level. Any help here?

Also, the 4-level grayscale routine itself was kind of slow.  I have changed that now for the next update and also the 3 level routine so they are much faster and better looking than before.  I even managed to get 3 color gameboy gray (completely flickerless) with more than a few things happening at once.

Absolutely.  Also, the 4 level is more convenient since you no longer use the "Setup" command, it is automatically initialized if it detects it has not yet been initialized just like the 3 level.

Well, I put the for( after the pt-offs, but I'm still getting the checkerboarding. I also tried pt-on instead, but it looks exactly the same. It does flicker less though, so it seems it's running faster.

If it helps, here's the source file.