EMC acquired Pivotal Labs in 2012. Months later, VMware and EMC each spin out a division, which included Pivotal Labs from EMC, as a new private company called Pivotal Software.
Dell announced its acquisition of EMC in 2015 which completed about 11 months later, in 2016. EMC shareholders each receive fractional shares of a publicly trading tracking stock representing Dell-EMC's interest in VMware in addition to cash for the non-VMware parts of EMC.
In December 2018, Dell went public by buying back its interest in VMware from the shareholders of the tracking stock.
So what you said about VMware being the only public part of Dell computers was true, up until the end of last year. Dell as a whole has been a publicly traded company for nearly 8 months now.