Thought I would chime in, and say I'm pretty sure the first video is Aikido S.A. (Shoot Aikido (Shoot is sort of Japans MMA in some ways)).
Not 100% sure, but I've seen it referenced in context with A.S.A. and the dojos seem to be located at the same places.
My own, completely uneducated, view on it:
As long as you use the Aikido principles in your Atemi (and not just pure strength, for instance), I say it's still Aikido. Gozo Shioda talks quite a lot about the importance of Atemi in his semi-biographical books.
Sounds interesting. May have to look into that. How, in you opinion, can atemi follow Aiki principles?
I began my fascination with aikido exactly 3 years ago. I'm definitely not the authority on aikido, but I'll offer my own $.02. One thing that we must keep in mind as beginners is the vastness of O'Sensei's knowledge and teachings. Much of what he taught was based upon the idea that you actually understood and/or had a background in martial arts. Some of that has been taken for granted over the years. One thing that we try and stress in our classes is how important committed attacks are to the development of Aikido. Proper stance, proper punching technique, proper kicking technique is something that we try to be aware of while training. I'm fortunate in that many of the aikidoka that I train with, including our sensei, have vast martial arts, military, and law enforcement backgrounds.
Aikido was founded on principles found throughout karate, judo, aikijutsu...etc...all aggressive striking arts. As you progress throughout your aikido training you'll see that the strikes are there. If that's what needs to be done then it can be be done. Many people watching videos of O'Sensei later in his years wonder how a man can do the extraordinary things shown in those videos. Throwing people with, what looks like, nothing but a mere glance or simply his Ki. We forget that there's been years upon years upon years of hoaning his craft. He had to kick a lot of ass before he got to that point. The higher learning aspect of O'Sensei's Aiki(do) era teachings put a different twist on the possibilities of "the atemi". This is a highly debated topic throughout the Aikido community. My belief is that the use of atemi is of upmost importance. Whether you decide to pull the punch or not is up to you. I don't necessarily subscribe to the theory that Aikido is a completely painless art. That to me is the beauty of it. Anything...and I mean anything is possible with Aikido.
If you can go to a seminar with George Ledyard Sensei...or watch some of his videos. He does a great job of describing atemi and striking. He's one of the best.
I think that teaching commitment (too much) can backfire. A lot of Aikidoka have a false expectation of what an attack looks like because we generally fully commit to our attacks, which makes it easier for Nage. Without doing so, you'll likely expect resistance and have to work around it, which I think is important anyways. But for the sole purpose of training techniques, it is necessary to commit fully... just gotta make sure that the newer Aikidoka mistake that as what could happen for it will likely not, but I definitely agree with everything else.
Another post I enjoyed on the other forums was where a member said that in order for an Aikidoka to be effective, they must be in a position to strike their opponent down, but use technique as an alternative as to show compassion. They quoted their teachers as saying something along the lines of "if you want to be able to make the decision to be able not to destroy, you must first know how to destroy". I personally love that outlook. Very much like my philosophy on martial arts: If you want to truly know and appreciate peace, you must first understand war (or in this context, self-defense and intense levels of training).
Just my humble opinion though.