My understanding is that white ink isn't really a true ink, as inks are always transparent and so white ink must have something added to it to make it opaque. Like acrylic binder.
With that out of the way:
I add light values to pen & ink drawings in a number of ways: colored pencil, gouache with a brush, and lately I have been trying some white gel pens.
I want a white I can use with a dip pen and brush, and so have been hunting for something that clicks with me. (Gouache out of a tube won't work well with a dip pen nib.)
The top "clouds" were applied with a No. 2 Robert Simmons Expression synthetic round brush.
The linear marks were made with one of my all-time favorite and most-used nibs, an Esterbrook 988:
Esterbrook 988 in a Koh-I-Noor holder.
They all felt serviceable with brush and dip pen. I could make do with any of them if I had to. I think the Daler-Rowney flowed the best in the dip pen, And the cloud it produced has the most subtleties.
This drawing shows the effect of a dip pen putting in the light value lines: