Hello Lennard,
Sorry for answering you until now, could you please try using LPCXpresso 8.2.2 or even better, MCUXpresso 10.2.1? I´m going to ask you to please install the IDE outside of the Programs files folder and then, follow the Getting started manual one more time, please.
Once everything is correctly installed and ready, please build and run the blinky example without any modifications and, manually start the RedLink server (is a red icon in the toolbar, looks like a red chain that is tilted to a 45º angle).

After this, please share with us the output you are getting.
Thank you for your patience Lennard.
Best Regards,
Fabian