I felt that this was my definition of what being "cool" really meant.
I felt that, personally for me, this defined what being "cool" meant.
ご質問ありがとうございます。
① "I felt that this was my definition"＝「私はこれが私の（～についての）定義だと感じた」
"of what being 'cool' really meant."＝「『かっこいい』とは本当にどういう意味かについて」
② "I felt that"＝「私は感じた」
"personally for me"＝「私、個人的には」
"this defined what being 'cool' meant."＝「これは『かっこいい』を定義していると。」
ご参考に！
アメリカ合衆国 
