I agree with most of the comments you've received and also want to add that this woman sounds a little western-Ukrainian to me, so her "e" did sound a little like "a" in "hat" or æ, but usually at least when I say it it is more like the English e in "red". So, as you see, both work since, as Vinnfred rightly said, we only have six vowels and don't care about all these details x)
So basically any decent pronunciation of "e" that distinguishes it from а, о, и, і and у is good enough :)
That depends on your English, too,— General American has a pretty high æ, which is quite close to "e" in "red" (really inconvenient for non-natives, since "man" and "men" differ mainly in length!) RP has these two sounds wider apart.
I'd say that Russian and Ukranian variation of /e/ vowel may include American "a" from "bad" somewhere on the border but does never reaches the standard British variation for that vowel (unless the speaker is fooling around or imitating a foreign accent).
They do for non-native speakers coming from languages with fewer vowels in this area. Then both might sound somewhere around "e" in "when" (more or less). The dialect of an English speaker also matters. For example, in this man by Slick or SeanMauch and this men by falconfling the vowels start out rather close to me. Then the one in "man" slides into a more open sound and is longer whereas the vowel in "men" is abrupt and stays roughly the same.
However, the pronunciation of man by enfield does not sound confusing to my ear. It is definitely an æ.
I open every skill, but don't go through the lessons and test it out instead. I am taking the course to look for mistakes and send reports :)
It's often harder for me to pass the skill than it is for someone who just started to learn the language here. Because my vocabulary is apparently much bigger and my translations are often not accepted