Is the intention to wake for suhoor sufficient as an intention for fasting in Ramadan? Or is it obligatory to make up the fast if one intended to fast a specific day (the sixth) and it turned out to be another day (the seventh)?

"Niyyah" (intention) overnight is a pillar of fasting according to the majority of scholars. It is simply the resolve to fast during any part of the night, such as having suhoor with the intention of gaining strength for the fast. If it crosses one's mind that they will fast tomorrow, then they have formed the intention. There is no requirement to specify the day of Ramadan; whoever knows that tomorrow is a day of Ramadan and intends to fast it has formed the intention. Therefore, there is no need to make up this day, and it does not affect the validity of your fast.