Digital BW, The Print

I'm having trouble understanding the quality difference between printers
that are 720dpi vs 1440 vs 2880.  If I understand correctly, when the
manufacturer says 1440 they mean 1440x720 not 1440x1440 and when they say
2880 they mean 2880x720 not 2880x2880.

If that is true, then I assume for a 1440 printer it is putting down a
dither pattern at a pitch of 1440 squirts per inch (sqpi) in one direction
and 720 sqpi in the other direction.  If this is true then isn't the 'real'
resolution of the printer 720 not 1440 - just like my 1200x2400 flatbed
scanner has a real resolution of 1200?

Can someone point out where I'm going astray.  Is the printers pitch
1440x1440? Is the real resolution 720, or am I missing somethin?

Thanks

Kevin Gulstene