について is 'regarding' or 'about', while にとって is 'for' or 'in the case of'.
They're not at all interchangeable.
「あまり」の検索結果 - Yahoo!辞書
According to the Shogakukan Progressive dictionary (yahoo), apparently you can achieve the same effect as your sentence with a slightly roundabout phrasing using について、
私について言えばゴルフはあまり好きではありません
As for me, I don't care much for golf.
The same English would be a fair translation for, 私にとってゴルフはあまり好きではありません, but in the について case you're actually saying 'If (we're) talking about me, I don't like golf very much.'
(あんまり is of course just a sound change of あまり.)
jisho.org (or more accurately, the EDICT database used by jisho, wwwjdic, rikai plugins, etc.) isn't -wrong-, but it's trying to say the same thing that I said in what I think are confusing terms. It means to say あまり・あんまり + a negative sentence = 'not very'. Not that あまり・あんまり means 'not very' by itself in a negative sentence.
Yahoo/Progressive has a couple examples for あまり,
「あまり」の検索結果 - Yahoo!辞書
There's nothing wrong with using a dictionary, but I find it helps to check with dictionaries that have examples (yahoo, or goo, or if you can afford it you can buy online access to Kenkyuusha's 和英大辞典) or else to check examples separately from ALC,
あまりの英訳|英辞郎 on the WEB:アルク or weblio
「あまり」に関連した英語例文の一覧 - Weblio英語例文検索 or tatoeba
Sentences with: あまり
Also, words like あまり・とって・ついて are usually covered as grammar points in any good Japanese textbook. I don't know what text you're learning from, but perhaps it needs replacing or supplementing with A Dictionary of Basic Japanese Grammar
Dictionary of Japanese Grammar Series : Grammar : Japanese Bookstore : The Japan Shop