The definitions you have cited make the words appear very similar.
Possibility tends to be used without reference to whether it is planned for or not. Contingency tends to be associated with deliberate plans.
Here are some examples that highlight this, together with short comments about the selected examples. I've taken the liberty of skipping one philosophical definition of contingency since you are interested only in "everyday use":
Possibility noun - ODO
1 A thing that may happen or be the case.
‘relegation remains a distinct possibility’
- 'relegation' might happen, but there isn't necessarily any preparation for it.
1.1 [mass noun] The state or fact of being possible; likelihood.
‘The Fenians always faced the possibility of being infiltrated by British spies.’
- Presumably the Fenians would have prepared for the possible infiltration, but all the sentence conveys is that it was always possible, not necessarily that they prepared for it.
1.2 A thing that may be chosen or done out of several possible alternatives.
‘one possibility is to allow all firms to participate’
- phrasing it this way means that it was considered, but it does not necessarily carry the idea that it was prepared for or acted on.
1.3 Unspecified qualities of a promising nature; potential.
‘the house was old but it had possibilities’
- The potential for the house was recognised, but on its own, the phrase doesn't convey that there was a plan to realise that potential.
Contingency noun - ODO
1 A future event or circumstance which is possible but cannot be predicted with certainty.
‘a detailed contract which attempts to provide for all possible contingencies’
- The contract is the deliberate planning in case the contingencies eventuate.
1.1 A provision for a possible event or circumstance.
‘stores were kept as a contingency against a blockade’
- The stores were kept deliberately 'just in case' of a blockade.
1.2 An incidental expense.
‘allow an extra fifteen per cent on the budget for contingencies’
- The extra money was deliberately budgeted for unforeseen expenses.
1.3 [mass noun] The absence of certainty in events.
‘The event was briefed, and every contingency was mapped out.’
- Mapping out the contingencies was part of the (deliberate) planning process.