Chotto…ちょっと means always ‘ a little bit’?

Why ちょっと is used frequently and casually? Is it the meaning of ‘ a little bit’ always?

1おかしを　ちょっと　食べました。

( I ate snacks a little bit)

2 ちょっとだけ　ほしいな。

( I want a bit)

3 おかしは　ちょっとしか　残っていない。

( There are only few snacks left)

Those are describing the ‘ amount’ and mean ‘ a little bit’.

1 ちょっと　いそがしくて、まだ　メールを　みていません。

( I haven’t check the email coz I have been busy)

2 ちょっと　子どもが　熱を　出しちゃって・・・。

(kid got a fever and…)

Those are not the amount but there seems to have some implying meanings. When ちょっと is used like above, the magic word　ちょっと makes the happenings sound small. However actually the things are big.

3 ちょっと　妻が　交通事故にあいまして、明日は　休みます。

( My wife got a car accident. I am going to take leave tomorrow)

Looking at #3 example, do you think the car accident is a small thing?

Of course no, the thing is big but by using ちょっと, the listener feels the things are small.

4 ちょっと　遅れます。　ちょっと　寝坊しまして。

( I will be late a bit, I have overslept)

If your team member call you to say that, do you feel and accept that being late is not the big deal?