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2. meaning: 例や、似ているものを挙げる表現

Construction:
A + のような + 名詞
A + のように + 動詞普通形
A + のように + い形容詞
A + のように + な形容詞

嘘のような本当の話なんですよ。
山田さんって太陽のような人だよね。

Indeed the same meaning but the constructions are slightly different. Feel free to use those constructions above on the other page.^^

ために shows the purpose for something (taken from grammar page):
「私は着替える為に家に帰った。」
I went home in order to change my clothes.

ように shows a goal/desired end - it is also a purpose, but you use ように when the purpose is not guaranteed.
かぜをひかないようにはやくねてください
Please go to sleep early in order to not (so you don't) catch a cold

Here, the purpose/goal is not catching the cold, although the action (going to bed early) cannot guarantee it. In other words, whether or not the subject will catch a cold is not entirely controlled by the subject.

I am in a rush, so this description may be a bit lacking, but I hope it helps!