There's generally a fair amount of flexibility in notice periods. As an employer, if I was obliged to give someone two weeks paid notice when terminating their employment, I would often give them three, just to be sure they didn't feel they weren't getting enough.
When leaving a job, the smart way to handle it is to tell a manager in person, then hand them a letter that includes the sentence "my last day at work will be Wed Aug 29th" (or whatever.) If the manager feels that is not enough notice, they can say so. Don't let them set your last day.
In Canada, you can't be forced to work and you can't lose entitlements because you didn't give "enough" notice. So you could just walk out of the building and not come back, and the only risk would be a bad reference later. I don't know if that is the employment law where you are. But whether it is or not, take control of your timeline yourself and propose or announce your end date, and let the employer react to it. Don't focus on number of days, focus on when you will last show up for work.
Once you've given your notice and all have agreed to it, don't change it. If I read your question correctly, they are asking at the last minute for you to stay another day or so. And you had earlier asked for a day off. Neither of you should be doing that. During a notice period, everyone is trying to get work finished up, cleared up, and passed on to the replacement. The leaving employee is getting ready for the new job. An extra day here or there, when there are just a few days left, will mess up the plans. If you are able to give them one extra day, good for you. But if you can't, you can't: you gave your notice and that is that, it is time to move on to the next thing.