Well,
The external fine trimming of the overall T.C. by changing the collector current from 20 to 200µA will change the -2mV/°C very lightly , I would estimate maybe from -2.0mV/°C to -2.1mV/°C.
This should mostly account for the variations of T.C. at different operating temperatures, either 23°C nominal for non stabilized use case, and about 45°C for an ovenized use case.
For this RefAmp, the T.C. of the zener and the transistors BE diode then have to match exactly in terms of these -2.0 ..-2.1mV/°C for the transistor and +2.0..+2.1mV/°C for the zener, which has about 6.4V.
When you convert these latter values for the zener to a relative T.C. = dUref/Uref, that gives a quite narrow window of +312 .. +328 ppm/°C, which the zener structure has to be trimmed for, otherwise you will not get an overall zero T.C.
Even if the collector current variation changes the T.C. of the transistors to a much higher degree, you will anyhow get a quite narrow span for the zener, due to its 13 times bigger voltage.
I don't know, how you can change a zeners T.C., but one parameter is its absolute voltage.. the 1N821..829 have nearly zero T.C @ 6.2V, and higer zener voltages will have more positive T.C.s.
The T.C. of these type of zeners vary greatly from 5ppm/°C (1N829) to 100ppm/°C (1N821), which is accomplished by selection only. Here you see, that precision zener diodes usually have a big manufacturing variation.
So LT had to invent something to precisely trim this T.C. of the buried zener to the desired value.
Frank