It is mostly unfortunate that the original evangelists of email decided to use the metaphor of the postal service to describe the new electronic message format. Instead of thinking of email as literally electronic post, think of it as bad PR.
An email message is not carried to your computer by little electronic ponies across the electric Wild West, nor is it trudged to your terminal by little electronic men and women wearing their little electronic blue shorts. Thankfully, there aren't electronic post offices at which you have to go and wait for hours while someone's electronic child wails behind you.
Other than sharing the word "mail" and serving to transport thoughts between a rhetor and audience, physical post and electronic messages are two wildly different things.
The word "email" is part of a living usage set. Just because Random House and Britannica can't keep up with how a word is used does not mean that such usage is wrong. Have a Google for a gent named Philip Gove (or have a read about him here) and you'll be introduced to the centuries-old conflict between prescriptive and descriptive dictionaries and their ability to stay relevant to living languages.
The noun "email," today, has a commonly accepted usage roughly framed as "a message transported electronically" Something more specific is of course necessary (involving mailserver protocols and all that) but this is what people mean.
You'd likely not want to throttle someone for saying "I sent three messages yesterday," because that's a common usage. Just as common usage has progressed from "Web site" to "web site" to "website," so to has it progressed from "E-mail" to "e-mail" to "email." The word itself no longer requires its original expansion because, frankly, the usage is common knowledge.
If such ruminations aren't adequate, however, you might instead rationalize it by telling yourself that such persons are using "a letter" as the metaphor for "an email," and therefore are in the clear when saying "letters"/"emails."
(To note, the above-linked essay is from my own hand. I'm not hit-trolling, just that it draws from some 18 other sources on the topic of Mr. Webster, Mr. Gove and the tendency for prescriptive dictionaries to become minimized and irrelevant.)