Learn Spanish (Latin America): A propósito, no entendí esa indirecta. - Speaking of which, I didn't understand that hint.

Yes, both are very common discourse markers meaning "by the way." However, a propósito is often used when the new thought is triggered by the current conversation ("speaking of which"), whereas por cierto can be used to introduce a completely unrelated change of subject.

I've seen a propósito used to mean "on purpose." How do I know which meaning applies here?

It depends entirely on the context and position in the sentence. When set off by a comma at the beginning of a sentence, it acts as a discourse marker meaning "by the way" or "speaking of which." When it follows an action verb (like lo hice a propósito), it means "on purpose."

What exactly is an indirecta? Is it the same as a clue?

Not quite. An indirecta is an insinuation, a veiled remark, or a "dropped hint," often with a slightly passive-aggressive or subtle social tone. If you are looking for a helpful hint or clue in a game or mystery, you would use the word pista instead.

Why is it entendí (preterite) and not entendía (imperfect)?

We use the preterite here because the speaker is treating their failure to understand as a single, completed event tied to the exact moment the hint was dropped. Using the imperfect (no entendía) would describe an ongoing state of confusion over a longer period of time.