This is simply called "vowel reduction" in English: акающие or аканье (coming from the verb акать) is "with vowel reduction (of unstressed о)", whilst окающие or оканье (from the verb окать) is "without vowel reduction (of unstressed о)".
There is also яканье, which is usually simply transliterated into yakanye because it is simply a different vowel reduction to /a/ instead of to the standard Russian /i/.
The секающие vs шекающие just refers to dialectal differences in unvoiced sibilant production (cf. sibboleth vs shibboleth from Judges 12:5-6). These are called "hissing" vs "hushing" sibilants depending on pedagogical tradition (e.g. for Russian and for Southern Evenki), but most phonologists would consider the first set "dental"/"alveolar" and the other set "alveolo-palatal", "palato-alveolar" or "retroflex" (etc.) depending on the place of articulation.
Apparently there is also хакающие, where a velar fricative is used. Again, these would just be described as having /x/ instead of /ʂ/ or /s/ where appropriate.