The story obviously has some errors and misunderstandings so it's difficult to tell what is accurate and what is not.

Fuel planning is required prior to takeoff and is part of the required dispatch release. The dispatcher prepares the release then the Captain reviews and approves it. If the Captain wants changes those changes are negotiated with the dispatcher. The flight can not depart until both the Captain and Dispatcher have agreed on the details of the release.

Here's an example of how fuel planning works:

FUBO 2+30 25,000 Time/Fuel burn off from takeoff to landing
ALTN 0+10 4,250 Time/Fuel to alternate (if required)
RSV 0+45 18,750 Time/Fuel for required 45 min reserve
--------------------
REQD 3+25 48,000 Time/Fuel required at takeoff
TAXI 0+30 3,000 Fuel for start and taxi out
EXTRA 0+10 4,250 Extra fuel beyond what is required
--------------------
RAMP 4+05 55,250 Fuel on board at engine start

Alternates are sometimes required based on the destination weather forecast. If they're not required then alternate fuel is not required to be carried. The Required fuel is the amount that must be on board at the beginning of the takeoff roll. This is why a flight that is close to taking off will sometimes have to return to the gate for more fuel. This is a hard cutoff. If you are one pound under the required fuel then you can't takeoff. Extra fuel is fuel above the minimum required fuel. This is the figure that is flexible based on the flight conditions. More extra fuel is carried when delays are more likely, less is carried when the weather is good and no delays are expected. Ramp fuel is what the fueler fills the airplane to prior to departure.

There are two types of low fuel declaration; minimum fuel and a fuel emergency.

Minimum fuel indicates that an aircraft's fuel supply has reached a state where, upon reaching the destination, it can accept little or no delay. This is not an emergency situation but merely indicates an emergency situation is possible should any undue delay occur.

An emergency due to low fuel is a distress or an urgency condition and implies the need for priority handling.

From the single report of questionable accuracy, I'm skeptical that an actual emergency was declared. A declaration of minimum fuel is more likely but we'd need some better sources to figure out what actually happened.