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@PearsonArtPhoto's answer links to the page Apollo 8, 21 – 28 December 1968 by Hamish Lindsay at honeysucklecreek.net from where I have quoted below.

Question: What exactly turned on the light indicating Apollo 8 was starting to fall towards the Moon? Was this a calculation based on elapsed time and inertial guidance, or did the Apollo computer use some doppler shift information from signals received from Earth?

Was there a separate subroutine in the Apollo 8's computer program to generate this trigger? Considering memory space was so precious it seems surprising for there to be a separate subroutine dedicated to processing data and triggering a signal just to turn on a light, as cool as that sounds.

EQUIGRAVISPHERE

Borman, Lovell and Anders were the first humans to leave the Earth’s gravity. They also never felt any physical change when the spacecraft slowed down to 3,578 kilometres per hour relative to Earth and crossed over into the Moon’s gravity field at 55:38:40 GET (0629:40 AEST). They were 326,415 kilometres from Earth and 62,598 kilometres from the Moon.

Capcom Mattingly, Welcome to the Moon’s sphere.

Borman, The Moon’s fair?

Mattingly, The Moon’s sphere – you’re in the influence.

Borman, That’s better than being under the influence.

Jack Garman, in the Guidance Support Room in Mission Control, remembered that moment,

“One of my anecdotal stories as they were coasting towards the Moon, which was a long and fairly boring period, a number of us decided to take a bet on exactly when a light would light up on our consoles. This light was connected to a telemetry reading from the on-board computers, that signalled when they determined that they were falling towards the Moon instead of rising away from the Earth, that is, when they had escaped from the Earth. …

When that light came on there was silence – it was a kind of dawning – we were witnessing the first time human beings were falling away from the Earth.”

The Apollo Guidance Computer used a state vector either centered at the Earth or the Moon. The switchover point is the the lunar sphere of influence, defined in the AGC as 64,373,760 meters (https://www.ibiblio.org/apollo/NARA-SW/R-577-sec5-rev4-5.6-end.pdf PDF page 127).

When in the idle program P00 the AGC will periodically check if it needs to update the stored state vector. When that happens it also checks if it needs to switch the center of its coordinate system. In that case it calculates a lunar centered state vector from its current Earth centered one together with a stored lunar ephemeris. Or vice versa. And to keep track of the currently used coordinate system origin the AGC has a flag (a single bit). The symbol name in the AGC code for this is MOONFLAG: https://www.ibiblio.org/apollo/listings/Colossus237/ERASABLE_ASSIGNMENTS.agc.html#4D4F4F4E464C4147

The flagwords are all downlinked via telemetry so what the flight controllers will have done in this case is to dedicate a light on their console for the state of this flag in the AGC. When it switched over to lunar sphere of influence (wouldn't have happened at exactly the radius given above, because the AGC only updates periodically) then mission control will have known immediately.