Ręka - hand or arm

I read that ręka can mean both hand and the whole arm, including the hand.  I wondered: in what situations would it be clear what was meant by ręka.

Daj mi rękę - I am told means: give me your hand.

Informally, the word łapa (like an animal's paw) is used in phrases like:

Trzymać coś w łapie    -    to hold something in the hand

Iść pod rękę    -    To walk arm in arm or hand in hand

Iść z kimś za rękę    -    I think also means walking either hand in hand or arm in arm.

So how do we differentiate?

How about the two different injuries inflicted in the scenes from two very different films:

Odciął mu rękę could be applied to both, but may suggest the whole arm.

Odciął mu dłoń should make it clear it is the hand.  Dłoń can mean palm or hand

Odciął mu ramię should make it clear the cut was at the shoulder.

Shoulder... now there is another thing that can also be unclear to English speakers...

See this post