I am not sure whether your comments about "satya" are to be limited to the
four satyas or apply to "satya" in general. But I think one will run into
difficulties if one wishes to define "satya" in general as either "truth" or
"reality". If one looks at the use of the word "satya" in the Indian
Buddhist and the wider general Indian philosophcal context, it would seem
that the semantic range of "satya" covers both "truth" and "reality". In
other words, "truth" and "reality" are virtually synonymous - if a thing is
real, then it is true and if a thing is true, then it is real. There are
some statements in which the word can best be rendered by "truth" and at
other times as "reality", but this is perhaps just a product of the
semantics of those words in English. In many cases, including that of the
four satyas, I find that "fact" would fit best and cover both meanings - if
it were not somewhat inelegant, one might want to translate "aarya-satya" as
"noble fact" (leaving aside here the question of relationship of "aarya" in
this context). Thus, texts I have worked on recently speak of a disciple who
recognizes the fact (= truth-reality) of suffering and so forth.
That "satya" (sacca) can also and often mean "truth" seems quite easy to
demonstrate. Consider the term used as the opposite to "satya" in such
compounds as "satya-vaadin" / "m.r.saa-vaadin" (sacca-vadin /
musaa-vaadin) - "one who speaks the truth" / "one who speaks falsehoods". Or
else the concept of "satya-vacana" - the efficacy of the utterance primarily
depends upon its verity, although its reality is perhaps implicit.
Finally, your claim may be true that "satya" is always and only understood
as "reality" in the Theravadin commentorial tradition, but one can infer
that this was a minority opinion in the greater context of Indian Buddhism
since both Tibetan and Chinese Buddhist translators over many centuries,
both inheriting considerable Indian Buddhist exegetical traditions, always
use a word (T: bden-pa / C: di) that means "truth" and not "reality".