Upgrade to El Capitan fails

I have a 2009 white MacBook, running MacOS X 10.8.6 Snow Leopard. I can upgrade it to High Sierra according to the Apple-website. I thought I could do this with Internet Recovery, but I have no way booting into it. When I try to download it on my new Mac and move it I get the notification: "Can't verify if this is a genuine Apple Installer" and the App Store says: "Can't connect to the App Store" when connected to both Ethernet and Wi-Fi. Can someone please help me getting El Capitan? I need that to be able to update to High Sierra. (I tried using the installer from the Apple Support website, but, the Installer always fails.)