just make sure you don't make any more mistakes
Chill bro, they've offered you the job now and you have the date for the induction. Keep in mind, there's no harm in asking for the date again because you forgot. Being honest is always good, but I wouldn't worry about it this time. No point analysing it any further, just go to induction and enjoy your new job.
They will give you a contract. You will have to work hard and build up a good reputatuion. Forget about it and move on.