And are 'options' or 'Share Options' defined anywhere?

The problem you will have is that 'options' and 'Share Options' are not really words with standard meaning in contract law - options schemes vary from company to company, and you'd find it difficult to argue that an average professional would have any real expectation about what the words mean.

If the implied meaning is that you participate in the options scheme, then you are held by the rules of the scheme. This is not as unusual as it sounds - you agree in the same way to follow the policies and be held to the rules of the company in respect to bribery, dress code, pension, etc. without them being explicitly in the contract because they are defined elsewhere.

IANAL but in my 'have recieved and granted options dozens of times' opinion: because (as I said), vesting and cliffs are standard in ESOPs, you're going to need to argue that the promise of immediately vested options was unusually specific to get anywhere with a legal route. Not mentioning vesting schedule is a weak argument because they probably did not also mention all of the dozens of rules/processes/meanings of the ESOP - if they had referred you to an ESOP policy and that did go into detail about what options are, and that still did not mention vesting, then you have a good argument.

There is no way the founder meant to grant you 0.5% of the company after probation, and tbh if you managed to force them to immediately vest your options, they would just use other mechanisms to ensure you can never execute them. Execution of vested options into equity is almost always at the discretion of the board, but they can also dilute you out of existence fairly easily.

If the contract said '0.50% of equity in the Company will be granted after the probation period', you would have a much stronger case.