All purposes, period. Say an armor crystal can only go on light armor: does that fall under the category of “movement or other limitations”? Yes, it does: it’s a limitation on the armor’s ability to carry this armor crystal. Anything that is restricted to a particular weight class of armor is affected by mithral.
In effect, mithral breastplate is a light armor, not a medium one. Mithral full-plate is a medium armor, not a heavy one. The only time an armor doesn’t count as a weight class lower is when the armor was light to begin with (and thus it never counts as a weight class lower).
This consensus on this ruling is incredibly wide; I have literally never heard anyone suggest it be otherwise. The description of mithral is somewhat vague (it counts as lighter “for other purposes”) but ultimately we have a vague but exceedingly broad characterization, and then we have absolutely no suggestion anywhere of any kind of exception.
For what it’s worth, and noting the problems with the FAQ, the FAQ agrees:
Is a character proficient with light armor, such as a
rogue, considered to be proficient with mithral breastplate?
What about a character proficient with medium armor,
such as a barbarian—is he considered proficient with
mithral full plate armor?
The description of mithral on page 284 of the Dungeon
Master’s Guide is less precise than it could be in defining how
it interacts with armor proficiency rules. The simplest answer—
and the one that the Sage expects most players and DMs use—
is that mithral armor is treated as one category lighter for all
purposes, including proficiency. This isn’t exactly what the
Dungeon Master’s Guide says, but it’s a reasonable
interpretation of the intent of the rule (and it’s supported by a
number of precedents, including the descriptions of various
specific mithral armors described on page 220 of the Dungeon
Master’s Guide and a variety of NPC stat blocks).
Thus, a ranger or rogue could wear a mithral breastplate
without suffering a nonproficiency penalty (since it’s treated as
light armor), and each could use any ability dependent on
wearing light or no armor (such as evasion or the ranger’s
combat style). A barbarian could wear mithral full plate armor
without suffering a nonproficiency penalty (since it’s treated as
medium armor), and he could use any ability dependent on
wearing medium or lighter armor (such as fast movement).
The same would be true of any other special material that
uses the same or similar language as mithral (such as darkleaf,
on page 120 of the EBERRON Campaign Setting).
So on the one hand, we have
actual text, though it is somewhat vague,
every example to be found anywhere in the system, including NPC statblocks, descriptions of specific armors, and the actual presence of “mithral breastplate” listed under light armor in Races of the Wild,
the opinion of whoever was writing the FAQ, and
the extremely wide consensus of D&D 3.5e players.
On the other hand, we have... well, we have that the wording is kind of vague, so maybe it doesn’t actually cover some cases. Which, fine—something could easily say “this only works in light armor, and mithral medium armor doesn’t count.” The rules would allow such an exception. But no such exception is written anywhere in the system (and it would be dumb if it did).