As I already said, if you use different sounds in лес and весь, it is incorrect independent of what sounds you use. In the IPA samples you linked it seems that in [e] they do not pronounce a pure vowel, rather they include a part of a preceding soft consonsnt and in [ɛ] they do not. This way it seems to me that you cannot pronounce their [e] after a hard consonant and [ɛ] after a soft one. As I already said, softness is attributed to consonants in Russian, not to vowels.