I have a hard time making out the first picture, and my guess is that the clay looks thicker than it really in on the second picture (the light source is coming in obliquely which lengthens the shadows). Also, in the case of the Japanese sword, you're probably looking at an extremely fast water or brine quench compared to the fast oils many of us use.
I'm no hamon expert by any means, but I've played around with them for a while now, and I've tried to pick up hints from other makers. The clay is really just a tool used to control heat. That's what it's really all about. Clay on top of a part of the blade may or may not mean softer steel underneath. Don't focus on what's happening with the clay; focus on what's happening with the steel. If the steel has reached austenitizing temperature for a sufficient time period (depending on the steel) and it drops temperature fast enough during the quench, it will harden no matter what is on top of it. In a like manner, if the steel doesn't drop temperature fast enough during the quench, it will not harden no matter what is or is not on top of it.
So, just use the clay to "push" the heat around and to hold on to it for just a little bit. If the edge of you blade quenches quickly enough, ashi lines covering the edge won't prevent hardening. However, if the clay is too thick or the ashi lines hold too much heat at the edge, you may not get a fast enough quench to harden the steel because of all of the retained heat in the clay.
Again, it's something that you have to play with to get right, and the subtleties depend on all of the variables of your particular heat treating apparatus and the particular knife it's used on.
Don't you love vague answers

. My students know that my favorite answer is "it depends" because most of the time it does.
--nathan