E3 is high pressure protection. Refrigerant pressure climbed past the safe limit and the system shut down to protect the compressor. On a Carrier split the usual reason is that the condenser cannot get rid of heat, either because air cannot get through it or because something in the refrigerant circuit is holding pressure where it should not be.
A condenser with leaves, grass clippings or pet hair packed into the fins cannot reject heat, so head pressure rises until protection opens. The indoor filter does the same job in heating mode. Cleaning both is the cheapest test on the page and it resolves a good share of E3 codes on units that have not been serviced for a few years.
It reads backwards after a high pressure trip, but a system short of charge can run high head pressure because coil temperatures get out of step and the compressor works against them. This is why charge gets measured rather than assumed. A shortfall means a leak, and the leak gets found before any gas goes in.
The pressure sensor can fail and report a condition that is not there, shutting the system down on a false reading. This is the diagnosis of elimination: once the coils are clean and the charge is correct and the unit still trips, the sensor becomes the likely cause. Replacing it earlier is how a healthy system gets a new part for nothing.
The return filter is removed first, then the indoor coil is checked for dust matting and the outdoor coil for leaves and grass. Both are cleaned before gauges are connected. On E3 this is the most likely fix rather than just good practice, and it costs nothing but time.
Charge is confirmed by weight or by subcooling and superheat, not by a single pressure reading, because pressure moves with outdoor temperature. If the system is short, the leak gets located before any gas goes in. Topping up a leaking system delays the next E3 by a season at most.
With the system isolated, the sensor reading is compared against a gauge on the same port. A reading that does not track the gauge, or a switch that opens at the wrong pressure, has failed. This is how a false E3 gets caught before a part is ordered for a system that was fine.