What did astrologers quote?
Do you want whether दुर्मुख is the name of संवत्सर or दुमुखी is the daughter of Lord Hayagriva?
For the second question, all the search did not yield any positive result.
Durmukha is 13th year of 60 years:
६० वर्षमध्ये १३ वर्षभेदे “तुषधान्यक्षयो
देवि! सर्वशस्यमहार्घता । व्यवहाराश्च नश्यन्ति दुर्मुखे
दुर्मुखाः प्रजाः” ज्यो० त० भवि० पु०
Why the names are given cannot be answered and who named them?
You can find in astrology books explanations like the above. Each name can be explained in relationship to each year's events predictions.