Let me use the 3 level grayscale example.

For example you got the following sprite:

BBBBBBBB
BBBBBBBB
BBBBBBBB
GGGGGGGG
GGGGGGGG
GGGGGGGG
WWWWWWWW
WWWWWWWW

B=black, G=gray, W=white. For the Gs you would quickly swap them back and forth from white to black. However, because the 83+ LCD is not memory-mapped, flickering will occur. However, what we generally do is make the G part a checkered pattern that is inverted rapidly. An optical illusion will cause our eyes to not notice as much flicker. The flicker will happen, but you'll see some sort of checkered gray pattern that doesn't seems to flicker as much instead, even if each pixel really flickers as much.

For 4 level grayscale, instead of checkered patterns some sort of scanlines needs to be used, and in ASM there's another method involving interlacing

Here's an example screenshot attachment of 3 level grayscale with the LCD blurring eliminated completly

Actually, the 4 level grayscale is this pattern:

11011010
10110101
01101011
11010110
10101101
01011011
10110110
01101101

In the one you posted, there are the same number of 0s as 1s in the 3rd and 6th columns making those pixels medium gray instead of dark gray.  This pattern can also be inverted bitwise for light gray.

In any case, never use vertical or horizontal scanlines for grayscale. Firetrack2 by Benryves does the former, and the flicker is really noticeable.

The idea with interlacing is that in 1 byte you can show 75% of the data from the dark layer, and 25% of the light layer at the same time. So instead of showing 1 layer then the other, you can mix them into the same LCD update, which helps reduce flicker.

OMG, i never fully understood that before, but when you explain it that way it's crystal clear! Thanks tr1p1ea!

the xLIB method in BASIC is

11011011
10110110
01101101
11011011
10110110
01101101
11011011
10110110

for light gray it is
10010010
00100100
01001001
10010010
00100100
01001001
10010010
00100100

However, this takes more space since each layers are hard-coded into sprite/tiles data and if you have several plain gray tiles in a row in a map, the scanlines might look a little weird

I think one of the basic grayscale tutorial used 1/4, 2/4 and 3/4, for 5 level grayscale. For the 2/4 it was a normal 3 level gs checkered pattern

what i find as a very annoying dilemma is that with one byte, you can have up to 9 level grayscale:
00000000
00000001
00010001
00100101
01010101
11011010
11101110
11111110
11111111

If you wanted to conserve space and keep one pixel in 3 bits, i.e., 8 pixels in 3 bytes, you have to leave one of the levels out...so, which one?