They are both accurate. Given that palatalized consonants are pronounced with the middle part of your tongue raised, the vowels adjacent to them are also raised (if there is room to raise at all) or at least experience some glide near the border.
In essense, what you hear as an e might be the actual unstressed и after a somewhat more tense и right where it starts. There is significant allophony in Russian.
A cat is кошка. Words of Slavic origin (and most old loanwords) will end in а, я, о, е, ё or a consonant in the Nominative case (the last includes nouns that end in a consonant followed by ь).
Only indeclinable words of foreign origin can end in и in the Nominative singular—it turns out, the Russian word for "cat" is not one of them.