wa 'alaykum salam wa rahmatullah
The intention for salaah is in the heart akhi, so when you stand you know what prayer you are performing - realizing automatically that it is late - and that is your intention. I dont think it resides further than that, as you know a similar instance occured to the Prophet
and his companions, but he never mentioned anything specific which should be done with the intention - and assuredly he
did not say it outloud. The following is the hadeeth:
Al-Bukhaari (570) and Muslim (681) narrated from Abu Qataadah (may Allaah be pleased with him) that the Prophet (peace and blessings of Allaah be upon him) was with some of his companions on a journey, and they halted at the end of the night. He lay down his head, then he said: “Guard our prayer for us.” The first one to wake up was the Messenger of Allaah (peace and blessings of Allaah be upon him). When the sun was on his back. We woke up, startled, then he said, “Ride on.” So we rode on and traveled until the sun had risen, then he stopped and called for the water vessel I had with me, in which there was a little water. He did wudoo’ from it, using less water than usual. A little water was left in it, and he said to Abu Qataadah, “Guard your water vessel, for you will see something happen with it.” Then Bilaal gave the call to prayer and the Messenger of Allaah (peace and blessings of Allaah be upon him) prayed two rak’ahs, then he prayed Fajr, and did as he did every day. The Messenger of Allaah (peace and blessings of Allaah be upon him) rode on and we rode with him, and we began to whisper to one another, saying: What expiation is there for what we have done by neglecting our prayer? Then he said: “Do you not have an example in me?” Then he said: “There is no negligence in sleep, rather negligence is the fault of one who does not pray until the time for the next prayer comes. Whoever does that, let him pray when he remembers it, and if it is the following day, then let him pray at the time of the prayer (that he missed).”
Allaah knows best
wasalam
