Every step is a kick - every punch is a throw
Every kick is a step - every throw is a punch
A kick may act as a step if one puts the kicking leg down other than where it started. The kick can cover one as they close the gap for example. A step may act as a kick if the stepping leg is used to make contact with the opponent's leg, breaking his balance. And this is stretching the definitions of stepping/kicking. As such, the first quote IMO is overly broad at best, and mostly inaccurate. But it sounds very Mr. Miyagi-ish.
While in some TMA kata/forms a throw may
appear as a punch (hiding the true application), they are not the same kind of thing. No matter how I look at it, a throw uses pushing and/or pulling power with some part of the body being used as a fulcrum, most commonly a hip or thigh (there
may be a rare exception*). This means there is most always at least 2 contact points: The point of power transfer (grab or hook) and the fulcrum point. It's possible for 3 contact points if one is using 2 hands, such as grabbing the arm with one and hooking the body with the other while bending him over your thigh to the ground.
* Exception: If you have hold of the opponent's arm and pull one way while sweeping the leg in the opposite direction the opponent may go down. This is a scissoring action with no fulcrum, just bi-directional power transfers. I don't consider this a throw. I mention it just for clarification.
Aside from all the above, the type of power applied sets striking apart from throwing. Throwing involves follow-thru force (pull or push) to move the opponent thru space, often overcoming gravity. A momentary application of force such as a punch does not accomplish this (is a knockout a throw?). In the case of
fajin as shown in post #64, it can be seen force is applied for just a micro-second. Contact lasts less time than can be humanly observed. This is in contrast to what is observed in throws where contact is maintained for up to a very long half-second.
So, a punch is not a throw.