Articles are not mysterious once you understand their conversational purpose.
When speakers use the definite article, the, they are indicating to the listener that they have something particular in mind, and that they expect to listener to know what that thing is.
So, when Tom says to Ann "Can you play the video?" he is assuming that Ann knows which video he is referring to.
When Ann replies "Video? What video?" she is letting Tom know that she does not know which video he is referring to, and she is asking Tom for clarification. "Video? What video?" means "I have no idea which video you are referring to. Which one do you mean?"
If Tom is going to give Ann the clarification she requested, he needs to tell Ann which video he has in mind. The idiomatic way to do so is to say:
"The video we watched 10 minutes ago"
"The video we talked about over lunch, the one you said you had a copy of".
or some other response that identifies the video Tom has in mind.
A statement like "We watched a video 10 minutes ago." is simply not a response to Ann's request that Tom identify the video he has in mind. It might be construed as a mildly sarcastic response, chiding Ann for having already forgotten the video they watched just 10 minutes earlier, but in Ann's defense, who would ask to see a video they've just seen without using the word again???
Tom: Please play the video again.