Thank you both for reviewing the lesson and offering your clarifications! It must just be my lack of exposure to hearing the German alphabet enough. I was able to clearly differentiate between the "E" and "I" in the singing version of the alphabet, but I still have trouble with the spoken alphabet. Last month, I returned from a trip to Freiburg to visit a close friend who is a native German, and I had difficulty understanding her when she tried to help me by spelling words for me that I did not understand. I was unable to distinguish between "E" and "I" when she spelled the words out loud to me.
It seems to me, while listening both to her and the recording, that the spoken letter "E" sounds very much like the spoken letter "I"; in other words, to my ears, they both sound to me like the American English "Eee". However, in the singing lesson immediately after the spoken lesson, the "E" sounds more like I would think it should: A is "Ah", E is "Eh", I is "Eee" (transcribed the way the German sounds hit the ear of an American-English-speaking native). Apparently I'll have to listen many, many more times before I will be able to discern the difference when spoken.
However, since you both agree that the spoken lesson has these two vowels well-pronounced, I'll just have to keep listening until the difference becomes apparent, or until I realize that I will never be able to tell the difference!