Does reference voltage long term drift reset after a power cycle?
Sadly long term drift cannot be eliminated by a power cycle, which would make this a non-issue for many applications.
Regarding how Texas Instruments define their testing procedure I have found a more detailed description of the tests they were running on the REF34xx references. I'm of course not sure that their testing method did not change, but I'd guess that they perform similar tests for other reference products as well:

Long-Term Stability
One of the key parameters of the REF34xx
references is long-term stability. Typical characteristic expressed
as: curves shows the typical drift value for the REF34xx is 25 ppm
from 0 to 1000 hours. This parameter is characterized by measuring 32
units at regular intervals for a period of 1000 hours. It is important
to understand that long-term stability is not ensured by design and
that the output from the device may shift beyond the typical 25 ppm
specification at any time. For systems that require highly stable
output voltages over long periods of time, the designer should
consider burning in the devices prior to use to minimize the amount of
output drift exhibited by the reference over time

Source datasheet
So what they are doing is measung the reference at regular intervals, over 1000 hours. Likely shortly after production.
They also have a presentation on references where they share a slide on long term stability here (slide 8). Key take away is: most of the ageing happens during the first 1000 hours.

Some of the effects they claim to be responsible for this should also happen during storage &quot;Long term shifts are caused by curing of the molding compounds, metal migration, and diffusion.&quot;
Now you will also see an effect by soldering the reference onto your PCB (slide 7) and that is likely to change as well over time as internal stresses get released.
Elevated temperature storage after production but before calibration can help reduce the effect seen in the field.

So the 1000 hours were likely measured after manufacturing. This is a complete lifetime behavior of the device and not a cyclic behavior. It is not very well defined and depends on a lot of factors.
We have seen the PCB have a very large influence on long term stability on our devices caused by moisture absorption of the FR4.
Thermal hysteresis can easily be mistaken for long term stability if temperature changes slightly over long intervals.