Definition 2 of the Late Apex: Intentionally make your apex slightly later than the optimum apex.
Note that implicit in this definition is that there is a known optimum apex location (maybe considered to be the same for all cars, or maybe not) and the driver chooses, maybe for safety reasons, not to apex at that point, but apex a little bit later. This definition has nothing to do with the geometric center of a corner. The optimum apex might be before the center or after the center, but the “late” apex is later than where the optimum apex is considered to be.
Usually the instructor of a novice student (who won’t know where the optimum apex is located in any case) simply shows the student where this “late” apex is located and the student practices precisely hitting that exact spot.
Let’s assume we have a moderately powerful car that needs to late apex a geometrically perfect 90 degree corner per late apex definition 1 that we discussed earlier. Here’s our old Figure 2, repeated, to show us just such a situation, perhaps with the lateness of the apex slightly exaggerated at B’.

Now imagine John Wannagofast in his 505Hp Zed06 Corvette at his first track day. Johnny’s instructor takes one look as his gold chains, leather pants, loafers with no socks and sticker Hoosiers on the car and knows he’s in trouble. While the line and apex location shown in Figure 2 might be exactly the way the instructor would drive John’s car, what he’s worried about is that John will not be very precise in his driving and may, in a testosterone-fueled frenzy brought on by being forced to give a point-by to a Miata with front fenders in two different colors, enter Turn 3 (let’s call it) too fast and apex too early. This is the driving line shown below in Figure 9 where I’ve drawn an arrow to indicate where, by accident, a novice might apex. Once that has happened there’s almost no way the car, in the hands of said novice, doesn’t leave the paved surface, maybe even spinning as it disappears down an embankment in search of a tire wall.

Here’s what happens instead: the instructor gets in his car and zips out to turn 3 and plops a cone down on the inner edge right at B” where I’ve shown the blue square in Figure 10, below.

The instructor now tells John, as they circle the course in the parade laps, that the way to take this corner is to get slowed down nice and early, turn in smartly as you come off the brakes, follow an arc around to the cone and then add throttle. About the 3rd session of the day the instructor may even suggest trying to get on the gas early, even before the cone is reached. Just make sure to get close to the cone and then allow the car to track-out to the left edge as you accelerate down the ensuing straight to be set up for the next corner.
This is the longer-dashed and bolded driving line shown in Figure 10.
Problem fixed.
John is now highly unlikely to go agricultural at turn 3. The instructor has built in a margin of safety while giving John plenty of good stuff to work on. Some people used to think (some still do) that this line, by virtue of allowing the start of acceleration to be earlier than the apex, will reduce the lap time because the average speed in the following straight is faster. They were wrong, but that’s another subject.