To me both words are interchangeable. Norbert, I think the only problem was that you misread the Pons-Collins abbreviation. It means 'housebroken' *is* a US form, and that it corresponds to 'house-trained.' The colon after 'US' is the key. (-:
Where I part company with Pons-Collins is over the hyphen. If it's not a BE form anyway, why introduce an unnecessary BE spelling? Apparently 'house-broken' has already been deleted from LEO, or was never in, which is just as well.
Where I part company with the remaining LEO entries is over 'house trained' (needs a hyphen) and 'domesticate.' You tame or domesticate a wild animal (of any age); you housebreak or house-train a puppy or kitten. Figuratively housebreak can also mean more or less 'civilize,' but that's not really the same as domesticate, nor does it really seem to merit a separate entry.
Re the context of small children:
'House-train' here sounds as odd to me as 'sauber machen' apparently sounds to German speakers from some regions. It's explicitly marked [Brit.] in Oxford-Duden, which offers these translations:
house-trained = sauber <Kleinkind>
house-training = (small child) Sauberkeitsgewöhnung
For toddlers AE has instead 'toilet-train' and [coll.] 'potty-train.'
Norbert, can I ask you a favor? Instead of just listing dictionary.reference.com, could you mention the actual title of each source? That site is a particular pet peeve of mine, since it includes not only AHD 4th ed. which is of course fine, but also, more unfortunately (at least the last time I checked) Webster's 1913 under some false modern copyright date, and Wordnet which is not really a published dictionary at all AFAIK.