diH2O, dH2O, and DI H2O. What do they mean?
I don't think there is a commonly accepted standard. If I had to guess, I'd say more often than not $\mathrm{di}\ce{H2O}$ is used to mean deionized water, but I've nothing but my own experience to base that on. Sorry for the lack of references. Also, $\mathrm{dd}\ce{H2O}$ is a common abbreviation that used to mean "double-distilled water" but now is used to refer to highly pure water produced by laboratory purification systems such as those from Milli-Q or similar.


$\mathrm{DI}\ce{H2O}$ = deionized water  
$\mathrm{di}\ce{H2O}$ = deionized water


Such water is often available from centralized water purification systems in buildings that host many chemical laboratories.  It isn't as pure as "$\mathrm{dd}\ce{H2O}$" (see below) but is subtanstially more pure than tap water.

$\mathrm{d}\ce{H2O}$ = distilled water 
$\mathrm{dd}\ce{H2O}$ = doubly distilled water
$\mathrm{mq}\ce{H2O}$ = Milli-Q water


The brand name of a popular type of water purifier that gives purity similar to $\mathrm{dd}\ce{H2O}$