What is the difference between 'peak oil' and 'oil supply crunch'?

'Oil supply crunch' means the supply is not enough to meet demand. It still allows the possibility that supply is increased to match demand. Peak oil means supplies can never be increased again. Governments prefer 'supply crunch' which sounds less alarming, especially since everyone assumes that this is because of growth in China. This in turn is accepted as we all enjoy cheap imports from there.

You are confusing terms here big time:

'Oil supply crunch' means the supply is not enough to meet demand.

To meet demand at what price? You are missing the price variable. Maybe this is the current price? But then for some people $5/barrel could be the 'fair price' for oil. My interpretation to 'Oil supply crunch' would be that oil production is either falling or not rising enough to prevent constantly rising prices. And again such definition is assuming that there are no structural changes to the demand side - e.g. recession.

Peak oil means supplies can never be increased again

There still might be some year/year increases, but the general trend would be down. These increases will be the primary reason we will be able to detect PO only when we are well past it.