Critical Fuel Scenario (CFS) is as you described, pretty much, but "half-way point" is really called Equal Time Point. It's based on loss of pressurization at (or near) ETP.
From the ATA 737 Nav manual:
As above plus flying at 10000 at ECON.
Approach the ETOPS alternate (almost always ILO for HI) at 1500', hold for 15 minutes, initiate an approach, go miss, execute a second approach, then land.
Now there's more. Various "bias" are added.
1. 5% bias for possible wind forecast errors.
2. 5% bias for anti-ice bleed use. (forecast icing, not very often going to HI)
3. 5% bias for ice accretion on unheated surfaces.
ATA and others have or can get relief from icing penalties Ops Spec B050, for CEPAC ops.
So, if you take the CEPAC relief you only have to add 10% bias to the CFS fuel scenario.
When I flew to HI with ATA on the B737, the arrival fuel using the above bias, could result in landing at your alternate with 1000 lbs. of fuel or less!