Sparsely populated, yes. "Barren and unpopulated," no.
By way of example, Florida in 1860 had a free population of fewer than 80,000.
Colorado Territory - 34,300
Nebraska Territory - 28,800 (statehood in 1867)
New Mexico Territory - 95,500 (comprising New Mexico and Arizona, separate Arizona Territory created by Congress in 1863)
Utah Territory - 40,000 (parts of Colorado and Wyoming)
Dakota Territory - 4,800 (comprising N. and S. Dakota, parts of Wyoming and Montana)
Nevada Territory - 6,900 (from Utah territory in 1861 during silver rush - statehood in 1864)
Washington Territory - 11,500 (included Idaho and parts of Wyoming)
You can see that your "unpopulated" statement is bunk.
A couple thousand people in a few tiny outposts scattered across the land equivalent of over 800,000 square miles is unpopulated by any reasonable definition. Even assuming your unsourced population figures are accurate (and I'm not at all inclined to believe that they are without a source given your lengthy history of fraudulently fabricating court decisions, quotations, and even civil war battles to suit your argument), the population density alone is virtually negligable so as to render a description of "unpopulated" accurate. To make a semantic dispute out of it also indicates your affliction with a severe anal retentive disorder, and one that has been festering for quite some time.