"A phase" is usually one of the three 'live' conductors - and a 'leg' is presumably just slang for that. A 'line' would also be the same thing. So no, I don't think you do have it right.
In the UK they've often been referred to by colour ("Red phase", "Blue phase", "Yellow phase"), though now our cable colours have changed, I guess they're "Brown Phase", "Black Phase" and "Grey Phase". They're also often labelled L1,L2 and L3. Custom and practice this sort of stuff does rather vary by country, so if you want to blend in with local electricians you'll probably need local advice.
'Phase voltage' is the voltage between one of the phases and neutral.
'Line voltage' is the voltage between two of the phases - usually sqrt(3) * phase voltage.
The word 'phase' comes about because the three live conductors carry the same waveform but shifted by a phase angle of 120 degrees between each conductor - in normal 3ph systems, one almost never talks/cares about the phase angle, because it's always 120 degrees and it's irrelevant to everything which electricians do in normal work.