Why does L freak out when Misa mentions the eyes in the tape recording?

I've seen a post from a long time ago explaining that he freaks out once the word "Shinigami" is mentioned because it throws a wrench into his logical-thinking brain. Though if you look closely, he initially starts to panic the second the word "eyes" is mentioned, not Shinigami. Why is this?

If L were to know about the eyes previously to watching the tape, then he would have been already aware of the Death Note and its mechanics. Though, this is impossible for a human to have already known. The entirety of society in Death Note has a warped view of Shinigami, believing them to be mythological soul-sucking demons. They wouldn't know anything about the intricacies of the Death Note, Shinigami, or the eye deal. There should be no possible way that L had known about the eyes, yet it seems like he did.