What is the LSST now? Where does LSST end and Vera C. Rubin Observatory begin?
The LSST Project is in charge of the construction project; building the telescope, camera and observatory and data pipelines and act as the "prime contractor". This includes designing and setting up the data processing and data distribution but the Project's job is essentially "done" once things start operating and the observatory transitions into regular operations. Once the Project delivers the observatory and it's signed off as complete, then the day-to-day operations transitions over to the NSF’s National Optical-Infrared Astronomy Research Laboratory. They will manage LSST along with other ground-based assets such as the (former) NOAO telescopes and the two Gemini telescopes. They will be the ones in charge of making the survey actually happen and keep functioning.

The Project was setup by the LSST Corporation LSSTC website which represents the institution and international partners, works with the Science Collaborations and is generally in charge/supports getting the science out of the observatory.

So as per the press release, the now named Vera C. Rubin Observatory consists of the mountain top facility in Chile which contains the 8.4-m telescope, now named Simonyi Survey Telescope after an early donor who got the primary work started, the commissioning camera (ComCam) and the main 3.2 Gpix DOE LSST Camera, along with Auxiliary Telescope and its dome and the headquarters in Tucson, AZ. This will be carrying out the 10yr sky survey, now backronymed to be Legacy Survey of Space and Time (LSST) to keep the same acronym.