This command should reboot the phone to Fastboot Mode (aka bootloader), and it seems that in the end you are getting there.
Edit: I don’t get an extra screen when I do this, I get Fastboot Mode directly. Perhaps connected to the following? …
Known fault of the fastboot command some Linux distributions would install for you, and observed on macOS, too.
Please make sure to use the current Android SDK Platform Tools.
On Windows that will output fastboot: usage: no command … fastboot needs a command apart from itself to know what to do.