4-4 point 3-3 invasion joseki, hane and inside hane
The inside hane at is indeed a third possibility to consider in the 4-4 point 3-3 invasion joseki (after and ). The order of and is moot, although the order given here gives slightly more room for variation: Black can exchange black a for white b before blocking at , and white can play at , giving up prospects in the center in exchange for more aji on the upper side.
With this result White has sente but Black has been able to shut off both sides.
Summing up on this idea: White has lived quickly, and left some outside aji by playing . Black's influence is going to work well with a stone at any of a, b, c and Black has successfully avoided the variation in which White makes a ponnuki on the upper side. Black has become quite thick.
Temporary Note?: Is Black thick, or does White have aji? The presentation is a little awkward, if not contradictory.
tapir: Can't it be a bit of both? White has a cutting stone she can put into motion later -> some funny business, Black is quite strong on both sides even then. In professional games it is mostly seen with a black extension already in place on both sides.
White can resist black's plans by playing the atari at here, then continuing with here or at a. However, is a big move in either variation, denying white's group eyeshape in the corner and thus keeping it weak.