I worked in the DoD when Gramm - Rudman (also referred to as Gramm - Rudman - Hollings, but less often so). It was the first time the introduction of the "sequester" came about as far as I know. Basically, a sequester is a cancellation or clawing back of money that had already been appropriated by Congress. This is a bit in the weeds, but there is a difference between Appropriated, Authorized, Committed, Obligated, and Expended in terms of Federal money.
Here is the budget process in very simplified form (apologize in advance that they have Hussein's likeness on it):
Congress has to appropriate the money to be spend for a specific, given purpose (this is very important). This is part of the "power of the purse" that we hear so much about.
The Dept of Defense, or the Dept of Homeland Security, or the FBI may be authorized to spend a specific amount of money, but unless the funds are appropriated, they cannot spend that money.
Each agency or department is authorized to spend once Congress passes the legislation and the President signs it into law. The omnibus did this, in a massive bill that included several appropriations bills.
Let's say the DoD wants to pave a runway at Ramstein Air Base in Germany, and it is going to use a German contractor to do so. The funds for this had already been submitted in the President's Budget, they were appropriated, and then authorized when the President signed the bill. The funds are then reserved, or Committed. This is a pre-Obligation of the funds. Obligated means that those funds have been assigned to be spent, and are no longer available. Finally, when the contractor gets an electronic transfer or a paper check, the funds are Expended.
What a Budget Sequester does implement automatic spending "cuts." Monies that had already been authorized cannot be Committed, Obligated, or Expended.
While Gramm Rudman was the first time this was used, the Sequester was later implemented as part of the Budget Control Act back in 2011. That Sequestration was the same concept, but legally different from any previous legislation. That Sequestration that we hear so much about was terrible legislation for one main reason: it didn't touch mandatory spending (which is 2/3rds of all spending), only discretionary spending, AND almost 50% of all cuts had to come from one agency - the Department of Defense, despite it being only 15% or so of all spending and less than (at the time) 3% of GDP spending.
Keep in mind, when the Budget Control Act came about because Congress refused to enact cuts with the "Super Committee" and brought us to the "Fiscal Cliff" back in 2013, these Sequester cuts came about in the middle of the Fiscal Year. That is why agencies such as the DoD had to furlough vital civilians who were performing maintenance on aircraft, ships and all the rest (engineers, teachers, DoD doctors and nurses, etc). It was a mess. Those Sequestration cuts were for TEN YEARS (until 2023), and the DoD has been living under this cloud. This is why President Trump has been trying to re-fund the military (part of the reason anyway).
Now, having said all of that, what does that have to do with what you both posted?
It looks like Trump is trying to use the existing Gramm-Rudman law and also a National Emergency to claw back funds that have already been Appropriated and Authorized (now that he himself signed the bill).
In layman's terms, Congress gave a big FU to Trump regarding the wall, Planned Parenthood, billions of Welfare giveaways, etc. - and Trump is attempting to give them a big FU in return.
I don't know where this leads. I will tell you that Gramm-Rudman failed in its stated goal (reducing deficits). Deficits exploded since 1985, as has the debt. It was a pinprick.
The Budget Control Act only succeeded in further crippling out military. Sequestration was only another nail in the coffin, because Obama had previously cut the DoD by massive amounts even before Sequestration. The "Stupid Party" (GOP) in Congress agreed to this "Super Committee" which was composed of Democrats and Republicans, and there was NO WAY the Democrats were going to allow cuts to Mandatory spending. So, the DoD took it in the shorts.
Perhaps this combination of a "National Emergency" and using Gramm-Rudman is a weapon Trump can wield. But, we are in uncharted territory here.