Defence of Usamah bin Zayd (may Allah be pleased with him)

It is narrated from Sayyiduna Usamah bin Zayd (may Allah be pleased with him) that he said: The Messenger of Allah (peace and blessings be upon him) sent me on a sariyyah (military expedition) together with some of his Companions. A man of the Ansar and I rushed towards a man of the enemy. I attacked him, and when I came near him he said 'Allahu Akbar', but I struck him with my spear and killed him, for I thought he had said these words only to save his life. When we returned, he (the Ansari) reached the Prophet (peace and blessings be upon him) before me and said: O Messenger of Allah! There is no horseman better than this horseman of yours. We pursued a man, and I was just about to reach him when he said 'Allahu Akbar', yet that did not stop him (Usamah) from killing him. The Prophet (peace and blessings be upon him) said: "O Usamah! What is this you did today?" I said: I attacked a man and he said 'Allahu Akbar'; I thought he had done so only to save his life, so I killed him. He (peace and blessings be upon him) said: "After he had said 'Allahu Akbar', what then will become of you (before Allah)? Why did you not split open his heart and look, so that you would know what he had said (sincerely)?!" He (peace and blessings be upon him) kept saying this to me that whole day. (Usamah, may Allah be pleased with him, says:) So now I will not fight any man who says 'Allahu Akbar', because the Messenger of Allah (peace and blessings be upon him) has forbidden me from it, until I meet him (peace and blessings be upon him).

The same narration, with the same meaning, is also narrated from Sayyiduna Usamah bin Zayd (may Allah be pleased with him) through another chain.
And as for what has been mentioned of Sa'd bin Abi Waqqas keeping away from the fighting (the reality of it is as follows):