(written from a Production point of view)
CBS Studios, officially known as CBS Broadcasting, Inc. and better known simply as CBS (formerly the Columbia Broadcasting System, est. 1927), is the television broadcasting network whose current holding company, CBS Corporation, owns the trademarks of Star Trek and all related marks, though the conglomerate has appointed the broadcaster as the titular copyright holder. As such, "CBS Studios, Inc." is therefore mentioned as the (co-)copyright holder on all Star Trek related merchandise released after 2005, bar the home media format releases of the Star Trek films, which only bear a Paramount Pictures copyright.
CBS's first involvement with Star Trek had actually already occurred in late April 1964, during Desilu Studios' attempts to sell Gene Roddenberry's Star Trek is...-pitch to a television network. On that occassion Roddenberry was accompanied by Desilu executive Oscar Katz, a former, longserving CBS executive. CBS was, after ABC earlier that month, the second outlet to which Desilu offered Roddenberry's Star Trek. CBS was in effect a natural choice for Desilu, having a first-refusal agreement with the company, as it was the longtime broadcaster that aired the Desilu television productions, including the I Love Lucy show from Desilu owner Lucille Ball herself. This actually made the first rapproachment with ABC something of a breach of contract, possibly explaining why Katz remembered the meeting as being "frosty", despite the longstanding relationship with both his employer and himself personally. CBS declined the proposition on this occassion, but not for the possibly perceived snubbing and not after both men – Roddenberry in particular – were thoroughly grilled on the offered property. As it turned out, CBS had its own science fiction show, Irwin Allen's Lost in Space, under development at that time, to be aired the next year, and never had any intention to add a second science fiction show to their line-up (ABC had even declined to receive the Desilu representatives for pretty much the same reasons, as it was about to air another Allen show a few months later that year, Voyage to the Bottom of the Sea). Both Katz and Roddenberry in particular, were convinced that CBS had picked their brains, after they had become aware of that fact. (Star Trek Creator: The Authorized Biography of Gene Roddenberry)
The show ultimately went to NBC, and just like Star Trek: Deep Space Nine and Babylon 5 would later become in theirs, Star Trek: The Original Series and Lost in Space became the science fiction franchise contenders of their day. CBS however, did pick up that other Desilu production, which was acquired by Desilu at almost the same time as Star Trek was, Bruce Geller's Mission: Impossible. At the time, CBS apparently treated Mission: Impossible better than NBC did the Original Series, which became somewhat of a bone of contention between the two Desilu production teams. An irked Associate Producer Edward K. Milkis recalled, "Another television show on the Desilu lot, Mission Impossible, was relegated a bigger budget than Star Trek, even though our show was the more difficult one to produce in our minds. The deal Desilu made with CBS was better than the deal they were able to make with NBC. NBC never understood Star Trek. They moved it around to different time slots in the three seasons it was on the air on three different nights." (Cinefantastique, Vol 27 #11, pp. 88-89) Another source of chagrin was that the original Mission: Impossible series picked up eight Emmy Awards out of twenty-three nominations, whereas Star Trek did none out of thirteen nominations. Albeit somewhat belatedly, Mission: Impossible would over time also evolve into a successful franchise of its own. (see also: Paramount Pictures: Footnotes)
When the original CBS Studios came under the ownership by the media conglomerate Viacom in 2000, it was on that occasion restructured and merged with Paramount Studios's television division "Paramount Television" (itself a 1967 merged version of Paramount's hitherto insignificant television division and Desilu Studios) and rechristened "CBS Paramount Television", with both broadcasting and television production activities concentrated under one roof. As such, the company laid claim to every episode of untill then televised Star Trek, including the by CBS originally declined Original Series, and was actually made responsible for the production of Star Trek: Enterprise.
In late 2005 the original Viacom conglomerate was spun off into two separate publicly traded companies: CBS Corporation and a new Viacom with the television division presently falling under the ownership of the CBS Corporation. On that occasion the former television division was split into two subsidiaries as well, the present broadcast television network "CBS" – essentially the original CBS Studios, and thus, in a manner of speaking, coming full circle – and the television production company as of 2009 known under its latest iteration "CBS Television Studios". The licensing and legalities surrounding copyright issues of consumer products related to Star Trek is presently controlled by the CBS Consumer Products sister division.
Unsurprisingly, it was the CBS broadcast network as CBS All Access that aired the U.S. premiere of the first episode of the CBS Television Studios produced Star Trek: Discovery series, "The Vulcan Hello".