What is the official name for Liore? I'm currently watching the FMA live action movie, which has Reole on its maps/subtitles. It seems as though this might also be true in the manga, at least according to this Wikia. However, Liore or Lior is the standard transliteration that I've seen for it, and I'm pretty sure that's what was used in the anime.

Both are technically (more or less) correct transliterations of リオール, but also somewhat unusual, given rs and ls are the same, and so it would usually either be Rior(u) or Liol(u). Reading from English, they also (at least to me) imply somewhat different stressing of syllables, Ree-ohl vs Lee-or.

Is there an official Hiromu Arakawa-approved way of writing it in English?
It's Reole.

Amestris is, canonically, an English-speaking country, and as such, all the locations are given "English" names (or, at least, names spelled out in the Latin alphabet used in English). This can be seen by looking at the original (untranslated) manga, which features a large number of character and place names written out in Latin script, rather than the katakana used for those names in dialogue. This includes the maps shown at various times in the manga.

Volume 17, page 70 offers one such example, featuring a clearly legible close-up of Liore/Reole on a map:



Additional context demonstrating that the image above is from the original Japanese language version of the manga:



Regardless of how the different parties involved may have translated it, it's quite clear that Arakawa's intended name for the city is Reole.