Presidential Proclamations and Executive Orders rely on similar authority, or lack thereof. A Proclamation is addressed to the world, and an order to the Executive Branch. Currently, the most likely question is to be about authority, and the exercise then is similar for both communications.
With Ceremonial Proclamations, like creating a special day of rememberance, the Presidents authority is basically just assumed.
When the Proclamation is supported by the Constitution, it should basically relate either to his job as Commander in Chief, example given, or his power to make treaties. This reminds me of our discussion on tariffs. I have not seen where the President has fallen back on Art. II Sec. 2 in that regard, but it would be easy to see how he might. The third basis noted is in my mind a BIG stretch. It is stuck in among a bunch of miscellaneous provisions, so that when I read it, I come away with, "Don't take bribes". But our current Supreme Court is fond of referring to this phrase as proof of how immensely large the Presidents job is. It has been used among other things as part of the argument for immunity.
When the proclamation is supported by statue, it is generally very clear that suthority is given. The problem is for what and based on what. The full text of 212(f) mentioned on the screen is "Whenever the President finds that the entry of any aliens or of any class of aliens into the United States would be detrimental to the interests of the United States, he may by proclamation, and for such period as he shall deem necessary, suspend the entry of all aliens or any class of aliens as immigrants or nonimmigrants, or impose on the entry of aliens any restrictions he may deem to be appropriate.” President Trump couldn't ask for more than that. This statute is still on the books! From 1952! The Presiident is given full authority to decide if things are detrimental to our interests, and full authority to do anything regarding the entry of aliens that he wants.