Misogi-no-Gyo

Well, I wouldn't say that either of these two definitions is correct. However, the statement that all throws are hip throws is correct. My take on this would be that when you reach a point when you let go of using strength (muscle) to down an opponent, there is a point at which the hips come into play. The key point here is that our center is the midpoint between our hips. When controlling an opponent, it is his center that we manipulate via whatever point they first offer out, be it a wrist, forearm, leg, fist, or what have you. This is a fairly intermediate level, say somewhere around 18 months worth of practice - when Go (square) becomes Jyu (circle). Of course Ryu is far off, say another 10 years, depending on the practitioner.

As one enters into the Ryu phase of training, the idea of "throwing" and opponent dies away. It is replaced with letting the opponent throw himself. Using one's own center to physically affect the position of the opponent's center. We used to have an expression that went, "Don't throw, move your center through the opponent's center."

Basically, the hips move in "complex" circles - complex referring to having multiple orientations with regards to the ground (i.e. containing a 0 degree component, a 90 degree component, ...etc.). If you could slow down an advanced practitioner, and focus solely on their hips, you would see that they turn their hips in relation to the uke (0 degrees) as well as drop their hips (90 degrees), along with another circle, entirely (somewhere in the neighborhood between 30 and 65 degrees).

This is the same for Koshinage, which literally means hip-throw. However, as Andrew pointed out, one has to turn his back in order to execute it.It is important to understand, that in Aikido we don't do this as a "choice" rather as a way of blending with a particular set of circumstances that that occurs because of the timing and proximity of the attack. In aikido, we do not have to grab the gi to perform koshinage. As a matter of fact, I can do koshinage without grabbing anything. As long as I can load you onto my hips, I can throw you, hands are not needed, although they can be helpful to ward off any incoming grab or punch.

In any case, when executed properly, one uses the third circle (mentioned above) as a method to enter (irimi) thus preventing the attacker from punching, kicking or grabbing with any relative power. The interruption of attacker's (uke) position (kuzushi), even for just a split second, is all that is needed, because the turning point, when the nage shows their back, occurs just after blending with the attack using irimi to obtain kuzushi in other words creating a moment outside of time and space (Aiki).

"Aiki" is a very advanced stage of practice. Everyone knows how it feels because they have had both moments when their teachers gave them such a feeling, and when they had moments of it during their own practice - those "Ah-Ha!" moments. Getting to the point when one can repeat it at will is what takes the average Aikidoka more than a decade to achieve. Of course, for others, more than a lifetime...