what is the name for this type of air intake gap on a residential roof which has no overhanging eaves and lacks soffit vents
In my neck of the woods, Mass, some builders call those "Hicks" vents. Not sure if that was a manufacturer or inventor or what.   I refer to them as drip edge vents.

I have these on my house,  I HATE THEM!  I'll tell you why.

On my house, unlike your diagram, there is a normal soffit and the drip edge and gutter are on the rafter tails.  More typical construction practice around here, puts the attic venting in a continuous soffit vent, a gap in the underside of the soffit, covered with a ventilated aluminum flashing.  I am slowly retrofitting them into my house.

My problems with the drip edge vent are two fold.  1) they were installed them poorly and in some places the water doesn't drip off the edge, but runs behind the gutter and down the siding.  I fixed this by retrofitting some more drip edge.

The more serious problem is in the winter.  If my gutter fills with snow and ice, additional water (melt or rain) will go up through the vents, and into the soffit and then down inside the wall.  I have had to put up gutter heaters and maintain them to prevent this.  And it caused multiple siding repaints before I figured out the extent of the moisture problems they are causing.  (not to mention carpenter ants)