Bill and others:
You are so right. But it gets more complicated .......... and fun, if you like digging and research.
The old Orange & Alexandria RR by the mid-1870s was then the Washington City, Virginia Midland & Great Southern RR or Virginia Midland for everyday conversation. The VM was then controlled by the B&O. By that time, the VM had very limited trackage rights and not enough trains were permitted by the C&O between Gordonsville & Charlottesville so Gordonsville was approached for a double-tracking expansion plan and they balked and refused the plan so the VM then created its own connecting link between the Charlottesville & Rapidan RR, directly connecting Orange & Charlottesville, originally a single-track line. This opened in December 1880-January 1881 and relegated the Gordonsville or Orange Branch into a lethargic jerkwater run, 9.05 miles each way. That continued for 10 years. During that time, the C&O made noises about wanting to build its own distinctly separate RR line to the growing DC region and that led to the lease Bill mentioned. The last VM jerkwater train ran on February 28, 1891 and for the next month, the C&O rebuilt the line with heavier rail and on April 1st, they commenced running into DC via the VM/R&D soon to be SR mainline.
The C&O also obtained trackage rights past AF tower in Alexandria and into the PRR's depot down at 6th & B (today's Constitution Ave), All that and much more are what led to the creation of Washington's Union Station and Potomac Yards over the next 15+ years and as they say ........ the rest is history.