If you're doing your hair like in the link you provided, you can definitely wash your hair with the extensions in, without messing them up. Same as you'd do with the extensions where the hair is braided all the way down.
Just don't forget to dilute your cleanser and rinse really thoroughly. You should separate the extensions from your own hair to comb it out, so as to avoid tangles.
Spinderella from Salt n' Pepa used to have these in her hair. She had them all over, not just for highlights the way you describe, and it took a lot of staring to realise they weren't actually her own hair. Since your own hair will probably be covering the point at which you braid them in, they should blend even better.
The site you provided the link to made it seem like a recent innovation, for a second, I almost thought they were talking about something totally new.
I even saw an English stylist on TV who claimed to have just invented this technique to prevent the breakage a lot of novice weave-wearers are experiencing from the glue & wefts