My first reaction upon reading this question, honestly, was to say to myself:
"Why would you use English 'translations' of the Japanese phrase for its grammar analysis?" (I said that in Japanese, though.)
「世話{せわ}になる」 vs. "to receive a favor" or "to be looked after"
"To receive a favor" is in active voice and "to be looked after" is in passive. To me at least, that should be a problem big enough to prevent you from a good and fair analysis of 「世話になる」, which is always in active voice within Japanese.
Notice that I am not questioning the accuracy of the two translations. They are actually OK. I am only saying that we should not be mislead by the coexistence of two grammatically different translations.
The super-literal translation of 「世話になる」 would be "to become trouble to/for". While you may never use that in actual translation from Japanese, but I feel it is imperative that you know that that is basically what the phrase means to Japanese-speakers. 「世話になる」 is not some idiomatic expression.
「A + は/が + B + の + 世話になる」 (B is the care-taker.)
= "A becomes B's trouble." (super-literal)
= "A receives a favor from B."
= "A is looked after by B."
Now, using 「世話をする」 instead of 「世話になる」,
「B + は/が + A + の + 世話をする」 (B is the care-taker)
= "B looks after A."
= "A receives a favor from B."
= "A is looked after by B."
Regardless of the situation/context, basically, you could express any "someone-helps-another" sentences using the two patterns above using 「なる」 and 「する」.
Let us look at your examples.
"Bob received a favour from Alice."
= 「ボブはアリスの世話になった。」
≒ 「アリスはボブの世話をした。」
"Bob helped out Alice."
= 「ボブはアリスの世話をした。」
≒ 「アリスはボブの世話になった。」
"Bob was looked after by Alice."
= 「ボブはアリスの世話になった。」
≒ 「アリスはボブの世話をした。」
TL;DR:
It is all about 「なる」vs.「する」.
「(Care-Receiver) + は/が + (Care-Provider) + の + 世話になる。」
「(Care-Provider) + は/が + (Care-Receiver) + の + 世話をする。」