Is a rak'ah considered missed as soon as the Imam begins to move to rise from rukoo' or upon his standing upright? And what is the ruling on someone who bows before the Imam says "Sami'a Allahu liman hamidah" if the Imam says it after rising from rukoo', and is it obligatory for the congregant to be precise in this matter?

The prayer unit (rak'ah) is counted if the latecomer (masbuq) joins the Imam while he is bowing, after saying the opening takbir (takbirat al-ihram), and shares with him in the sufficient extent of bowing, which is the bending where one can place their hands on their knees, and achieves a moment of tranquility with him, even if brief.

However, if this extent is not achieved, for example, if the worshipper descends to bow and the Imam rises, then this prayer unit is not counted, and the worshipper should follow him into prostration.

The worshipper must be certain or have a strong presumption that they have joined the Imam while he was bowing. If there is doubt about having caught the bowing, then that prayer unit is not counted.