This is quite confusing!
In the Standard IPA vowel chart, there are /e/ and /ɛ/. See the below picture:
However, many American English vowel charts don't have /e/. So, I think that some English dialects (like General American) don't have /e/, they only have /ɛ/. See the below General American Vowel chart:
In contrast, the British vowel chart has /e/ but doesn't have /ɛ/. See the below Received Pronunciation Vowel chart:
As you can see, the /e/ in British chart is in the position "mid-near front" whereas the /e/ in standard IPA is in the position "close-mid-near front".
The confusing things are in the dictionary. See the word "bed", some American pronounce it as /ˈbɛd/ (Source: learnersdictionary.com), other American dictionaries write it as /bed/ (Source: Cambridge Dictionary).
However, most dictionaries pronounce "save" as /seɪv/.
Also, most pronunciation teachers do not teach pronouncing /e/ as a monophthong, but rather they teach pronouncing it as /eɪ/. But it seems English native speakers cannot pronounce or have never pronounced the /e/ in the position "close-mid-near front" as in standard IPA. However, people using other languages can.
So, could you clarify /e/ and /ɛ/?