Yes, you are right. The the public directory is the root and you should first cd into it.
Please type the following in your command line:
DB_HOST="127.0.0.1" DB_USER="root" DB_PASS="tiresome" DB_NAME="timesheet_manager" php -d variables_order=EGPCS -S localhost:8000
or set up the public dir as a www dir in your server config.
see a screenshot:
ibb.co/gKbsHz