Whenever I read about options valuation, it's phrased in terms of payoff diagrams. For European options, the value at expiry is the difference between the strike and the market price of the underlying if I'm in the money.
However, the usual definition of an options contract is not a cash bet conditional on the market price of the underlying, but rather an offer to exchange cash for the underlying asset (or vice versa).
When I buy a call option in practice, is it common to receive the underlying upon exercise and then be required to sell it on the market to realise a profit? Or will the counterparty ever just hand over cash if I'm in the money?
Similarly, does a put option typically give me a literal right to sell the asset, or is it just a cash bet with the payoff conditional on the asset price?
What might influence the particular choice of logistics here?