I agree that 'stomachache' ('stomach ache,' 'stomach-ache') and 'bellyache' ('belly ache,' 'belly-ache') are absolutely synonymous, at least in colloquial AE, so that any LEO translation that uses one (e.g. Bauchschmerz/en, Magenschmerz/en) should probably also list the other(s).
For the abdomen in everyday language I use 'stomach' myself, not 'belly.' I think it's one of those words that varies regionally, like 'soda' and 'pop' and 'soft drink.' In this sense it may be exclusively AE, but BE speakers would have to say.
Stomachache, or bellyache, seems to be a term that originates in childhood (cf. 'tummyache') and usually describes a milder, more chronic symptom than I would imagine with an appendix. If appendicitis is more a pain (sharp, severe) than an ache (dull, medium to mild), it might more often be described as '(acute) abdominal pain,' or more colloquially, 'a sharp pain in the stomach/belly.' However, it might begin with a simple stomachache, who knows; I've (knock wood) never had it. <g>
BTW, if it weren't too much trouble, I might suggest taking this opportunity to add 'bellyache' (but not stomach- or tummy-) as a verb as well. It's a little milder than 'bitch,' a little stronger than 'complain.'
bellyache - (v.) [intrans.] complain noisily or persistently: 'heads of departments bellyaching about lack of resources.'
bellyache - (vi) (= complain) murren (about - über +acc.);
bellyaching - (n) (inf) Murren, Gemurre
bellyache - (v.i.) (sl.) jammern (about - über + Akk.);
bellyaching - (n.) (sl) Gejammer (ugs. abwertend)