Frequent short-term student visa - UK/US citizen

Your questions are...

[C]an I apply to the [short-term study] visa with my Jordanian passport rather than go with my US passport?

To answer this, we can turn to Paragraph 320, Subparagraph 7D-11, which says, in part (emphasis mine):

...and there are other aggravating circumstances, such as absconding, not meeting temporary admission/reporting restrictions or bail conditions, using an assumed identity or multiple identities, switching nationality, making frivolous applications or not complying with the re-documentation process.

They will not prosecute because you are a ‘small fry.’ They may not even impose a ban; however, they are likely to put a stop flag on both passports and ask both countries to keep them advised of new identities, new passports, and so on. It means they consider this to be an 'aggravating circumstance' and it's something they will not forget for a long time (if you are caught out that is). Note that if your school finds out, they will be obliged to report you to UKVI.

Will they know that it’s in my record that I’m not allowed to take another sts again?

No, there is nothing in your record. From what you wrote, the IO gave you a casual warning. There was no paperwork and you were not removed. There is nothing in your narrative to indicate that a ban (or any other restriction) was imposed.

Will they cause a fuss like last time?

We are not prophets, but I suspect the next time with your US passport will result in a removal, and with your Jordanian passport then Paragraph 320 is engaged.

Note that it is fine to for people to come and go on different passports. It is a normal event in a multinational society as we're in. You, however, would be using different passports to evade immigration control and this would be pretty clear to a caseworker who examined both passports. An IO would most likely be able to figure it out with a single passport, but that's merely supposition based on your narrative here.