You are still getting hung up on this...
You have been told repeatedly, by lots of people, that the label is just a teaching aid... and that these movements are much more than just what the label says. I get it that you won't take my word for it, or the word of anyone else here.
As I understand it, Funakoshi had a lot of influence in the creation of TKD, as well as being the founder of Shotokan karate. In the following article:
Lyoto Machida: Old-School Karate it discusses how he, Funakoshi, taught the "down block" as a throw. Not a block, followed by a throw, or a block followed by a grab, followed by a throw... he taught it literally, as a throw. So did his direct students.
I understand that you really don't agree with him. However, if I have to choose who to believe, Funakoshi, that founder of Shotokan Karate, which had a great influence on the creation of TKD... or some random TKD guy online... I choose Funakoshi's version, every time. But, if you think Funakoshi got it wrong... thats fine by me.
The letter "P" is just a letter. And it is always the letter "P." When you teach people what the letter "P" is you use words like "Push," "Pull," and "Pan." But, "P" can be used, as a "P" in lots of words... and not only as the first letter. It can be the end letter, as in "stop," or a middle letter as in "happen." It is still the letter "P." However, the letters in martial arts context, are the movements and principles, not the labels. You seem to be stuck with "P" being part of the word (label) "Push." Because "P" starts the word (label) "Push," thats what it is... And you are correct, you can use "Push" in all kinds of sentences and paragraphs to say lots of things. If you listen to Funakoshi, you should be able to find "P" in words like "Pull" and "Pan"... then soon in words like "Stop"... maybe eventually in words like "happen." (I can't wait for you to find it in "Phase")
Most of us, take Funakoshi to be correct, that these are movements are a lot more than their labels. But if you want to limit them to the label.... thats your choice. Just understand that the rest of us see a little more in these things.