grammar - Can ところで be ambiguous (time vs. location)?

I was studying the usage of ところ with particles and came across the following sentence and translation (linked at the bottom):

駅の近くでケーキを買ったところで彼女に偶然{ぐうぜん｝会った。

I met her by chance when I just bought some cakes near the station.

The article does not explain it very well but from what I gather (from other sources), ところで can be used to show circumstances surrounding the main clause. So the sentence above could perhaps be more elaborately (and robotically) understood as (please correct me if I'm wrong):

"I met her by chance. It happened when I (was out buying/just bought) a cake near the station."

My first question is, Is the translator's use of the word "just" justified? (Sorry, I couldn't resist the pun). I understand that 「～た形」＋ところ shows that the action happened in the very recent past and is commonly translated as "just" but is that meaning carried through when ところ is used as above? Stated another way, does the sentence imply that I just bought the cake (giving ところで a "time" nuance) or that I was out buying the cake (giving ところで a "circumstance" nuance) ?

Second, if I had seen this sentence out of the context of the article, I would have translated it:

I met her by chance at the place by the station where I bought the cake.

Is this also a valid translation or is there a reason that this can't be correct?

Article source

Answer

駅の近くでケーキを買ったところで彼女に偶然会った。

The sentence doesn't sound ambiguous to me. It means "I met her by chance right when I bought a cake near the station."

According to 明鏡国語辞典:

ところ【所(処)】〘名詞〙
⑥-㋒《「・・・(ようと)するところだ」「・・・ているところだ」「・・・(てしまっ)たところだ」などの形で、現在または現在に近い過去を表す語を伴って》動作が、直前・最中・直後にある意を表す。「今手紙を書いているところだ」「書き終わったところでベルが鳴った」

「～するところだ」「～ているところだ」「～たところだ」 indicate that an action is about to happen, is now happening, or has just happened, respectively. So the ～たところで in your example means "Right after (I) did~~" "Right when (I) did~~". For example:

• [夢の中で] 食べたところで目が覚めた。
  [In a dream] Right when I ate, I woke up.

Compare:

• 食べるところで (≂ 食べようとしたところで) 目が覚めた。
  When I was about to eat, I woke up.


• 食べているところで目が覚めた。
  
  While I was eating, I woke up.

---

I met her by chance at the place by the station where I bought the cake.

For the ところ to mean "place", you'd say:

駅の近くのケーキを買ったところで彼女に偶然会った。

×「～の近くで～した｛ところ・場所｝で」
⇒　○「～の近くの～した｛ところ・場所｝で」(avoiding repeating the locative case で)

So, if the sentence was 「ケーキを買ったところで彼女に偶然会った」 without 「駅の近くで」, then it would be ambiguous between "at the place I did..." and "right when I did..."