It's hard to standardize on the broader definition of single-tenant versus multi-tenant clouds because everyone seems to disagree on the definition "tenant" (naturally, I'll blame the marketing folks).
Everyone can agree that "tenancy" refers to resource isolation. My application does not know that your application exists. My application cannot steal your application's data.
Before we can define "tenancy" any further, we need to discuss scale. An application's environment could be as small as a single physical machine, as large as a rack, or even an entire datacenter. If you want to stretch things further, you could define an "environment" as an entire cloud, which in Rackspace's or Amazon's case, are multi-region, multi-datacenter behemoths. You could also draw lines between two environments using non-physical boundaries, like containers (LXC, Docker, Rocket), virtual machines, or virtual private networks (VPNs).
A public cloud is likely to be a multi-tenant environment starting at the scale of individual physical machines (several tenants will likely be co-located on the same physical box). This obviously implies that a public cloud is multitenant at any larger scale.
In a public cloud, a tenant is traditionally a customer paying a bill for resources they consume (hence the name: tenant). Whereas in a private cloud, one organization likely owns an entire datacenter (or at least a rack in the guest bedroom closet), and no one is being billed for the individual cloudy things happening on top of that hardware. But, that one organization might have 10 users running 100 projects in the cloud, and we refer to each of those projects as having tenant-level isolation from each other.
tl;dr Both public and private clouds are multi-tenant on a technical level, they just don't share the same business-focused definition of "tenant" (multiple paying customers).
We tried to avoid all this confusion in OpenStack by moving from using the term "tenant" to "project." Projects provide resource isolation, but we don't have a preconceived notion of who's footing the bill.