Be out at/for work

Hello Jandi

The final answers would come from native speakers, but let me try.

I think your two phrases basically mean the same: 'I am not at home to work'.

But "out" could mean "open about one's sexual orientation"
So someone might say : "She is out at work as well as with friends".

paco

In fact, this is my brother's question.
His teacher told him that
1. I'm out for work. - This means 'I'm going out to work', or 'I'm out of my workplace or home in order to do some work.'
2. I'm out at work. - This means 'I'm at my workplace, not my home.'

But I think that
1. I'm out for work. - No English.
2. I'm out at work. - I agree with her. This means 'I'm at my workplace, not my home.'
And I think we should say that
3. in order to mean 'I'm going out to work', 'I'm off for work'.
4. in order to mean 'I'm out of my workplace or home in order to do some work,' 'I'm out on business'.

"I'm out for work": to me it means "I'm (going) out (looking) for work"

Another possibility, which I didn't expect at all !!!

Thank you anyway, Paco and Pieanne!
Enjoy the spring flowers.

Yes!
Thank you, Pieanne!