There is nothing in that contract that says anything about 3 months notice period. The 3 months is the legal default for contracts that do not expire on their own, unlike yours, that has all properties of a limited time contract. I would personally see the detailed description of how you can end this contract as overriding any legal default.
But as always, with this specific contract in the original language, you need to see a lawyer to know for sure.
Your contract clearly states: you can leave your appartment whenever you want, even before the agreed upon time.
If you leave between the 15th and the end of a month, you have to pay for that month in full.
If you leave between the 1st and the 14th of the month, you have to pay the fair share of the rent for the days you where there. So for example, on a 30 day month if you lived there for 10 days, you still have to pay a third of the rent and the landlord will return the rest if you paid for the month in advance. If you live there for 16 days, you have to pay for the full month and nothing will be returned if you paid for the month in advance.
Please note that you need to "hand over" the vacated rental object during normal business hours. So don't go in there on the evening of the 14th at 16:59. And don't try to "hand it over" when you haven't moved your stuff out yet. At the hand over, you give the keys to the landlord and that is it, it is the last thing you do. Very likely your landlord will want to have a look at the rental object while you are there, so they can make sure it is all in order, you did not damage it or did not leave any of your stuff.
Generally speaking, there is nothing your landlord could do to you if you decide to leave early. They cannot make you leave even earlier or any other retaliatory shenanigans you may have heard of in other countries. In Germany, such contracts are not adversarial. You don't need to keep it a secret to the last second. If you know you want to leave on a certain date, inform your land lord, make an appointment for the "hand over" well in advance and save yourself (and them) all the stress from doing things last minute.