Thank you for responding so quickly!
I have previously backed up my phone with this computer, unfortunately it's been a while. So I don't have a current backup of my phone on my mac. So when I get prompted with "Welcome to your new iPhone" and whether or not to restore it from a backup or set it up as a new phone, should I proceed to set it up as new or will that delete the data currently on my phone? I don't want to restore it from a backup because the backup I have is so outdated.
As far as I understood your reply, setting it up as a new device is necessary, so that I can do a backup but then you wrote, that I ideally should have a backup before proceeding. But that is my whole issue: I can't make a backup because my phone wasn't recognized by iTunes. Now that it is, thanks to your first reply to the original thread question, I get prompted with the "setup from backup or as new" dilemma.
If it changes anything about your reply to my issue: I don't need to connect my phone to iTunes because of media syncing (as in music, films etc), but solely because I want to do backups of my phone.
Thank you for your help!