I suspect you're simply confusing yourself - there's no such thing as 'ground' on a portable scope (and in my case not on a mains scope either).
I presume all the button does is short the input to the common/ground connection, that's all. It's so you can adjust the reference to where you want on the screen, without been confused by any input. So connect the 'ground' lead to the chassis of the unit under test, and the probe to where you're measuring - then press the ground button and set the reference point on the screen.
All my mains powered scopes have a three way switch, AC, DC, and Ground - so in that case you set it to Ground to set the line in position, then back to DC (not AC) to measure the DC offset.