I want to use ubuntu/xenial64 box to create two separate VMs for two separate projects. I defined Vagrantfile in two separate project directories and added the line config.vm.box = "ubuntu/xenial64" to each.

The first box boots successfully. But when I do vagrant up for second project, I get the error

A VirtualBox machine with the name 'ubuntu-xenial-16.04-cloudimg' already exists.

In Vagrant's documentation it's clearly written that

Boxes are globally stored for the current user. Each project uses a box as an initial image to clone from, and never modifies the actual base image. This means that if you have two projects both using the hashicorp/precise64 box we just added, adding files in one guest machine will have no effect on the other machine.

Why then do I get this error?

I have already checked out other similar questions, but I don't understand their solution of deleting existing VMs that appear to have the same name. According to the Vagrant documentation quote above, that shouldn't be necessary. Am I missing something?