Normal trances reflect elves' connection to their primal soul, their current mortal life, and their past lives
Your quote from Mordenkainen's Tome of Foes leaves a lot of relevant information out. "Chapter 2: Elves" goes into quite a bit of detail about elves' trances.
Under the heading "The Elven Diaspora" (p. 36), it notes:
No matter where they are in the multiverse, elves of all sorts feel a special connection to the realm of Faerie, for it was their race’s first home after they were cast adrift. Even if they can’t name that realm or don’t know how to return there, vestigial memories of the place sometimes glimmer in their minds when they trance.
And under "Living in Reverie" (p. 37-38):
Perhaps more so than any other race, elves are familiar with all aspects of memory. From birth, elves don’t sleep but instead enter a trance when they need to rest. In this state, elves remain aware of their surroundings while immersing themselves in memories. What an elf remembers during this reverie depends largely on how long the elf has lived, and the events of the lives that the elf’s soul has experienced before.
It explains that during an elf's first few years, "the memories evoked during trance are drawn not from current life experiences, but from the fantastic past adventures of the elf’s immortal soul." It's not until adolescence (in the elf's second or third decade) that they first experience a memory of waking life during their trance.
They are considered adults towards the end of their first century of life, when they experience "the Drawing of the Veil": they no longer experience their primal memories, and only recall the events of their current life. As adults, they learn how to control the memories they experience during a trance, often choosing to recall useful or comforting ones as needed.
Finally, they experience another change and become "elder elves" in their third or fourth century:
At some point during adulthood, the reverie of an elf’s trance is first interrupted by a new form of unbidden thought. This seemingly errant memory arises not from the elf’s personal experience, nor from the memories of the elf’s primal soul, but comes from another life and another time.
As these intrusive thoughts become more frequent, they focus less on the world and their focus turns inward.
Regardless of how soon or how often elves experience such memories, most consider them a blessing from the gods. The experiences of other lives that are revisited during trance can be examined for lessons to be applied during one’s waking life, signs from the gods, or ways to open an elf’s perspective to other points of view.
A handful of elves in any generation never experience an other-life memory during trance. It’s hypothesized that these select few might be reincarnations of the original primal elves who sprang from Corellon’s blood and were allowed to stay in his company. Although most elder elves become more serene, these rare folk spend the rest of their lives throwing themselves into dangerous situations, as if daring death to try to take them.
The "Dreams from Beyond Memory" sidebar notes that all elves can sleep and dream as normal if they choose, but most surface elves avoid it; actual dreams, as in humans, are uncontrolled products of the subconscious, and most elves find them bizarre.
Towards the end of its life, an elf gets
cataracts in the shape of crescents, points down, that appear over the pupils of both eyes when the elf is in trance. This change, commonly known as Transcendence, is evidence that Sehanine Moonbow has opened the door to enable the elf’s soul to return to Arvandor — a direct sign from the gods that it’s time to get one’s affairs in order.
Drow can trance, though it's different in nature
As I mentioned, you've left out quite a significant portion of the "Drow Trance: Entering the Void" sidebar (p. 53, emphasis mine):
Drow enter trance just as other elves do, but they do not experience memories of a primal soul or of past lives. Often they recall nothing at all, but simply dwell for a time in darkness and silence, a respite from the dangers of their daily lives. When drow do dream, whether in trance or in sleep, they look for signs from Lolth or others of the Dark Seldarine. That drow do not experience trance the way other elves do lends credence to the idea that their souls do not reincarnate. Did Corellon forever bar the souls of dark elves from Arvandor and change them in some fundamental way? Or does Lolth somehow weave new souls for her followers, in the way that Moradin forges new spirits for dwarves? Only those entities know for certain.
The nature of a drow's trance seems to have nothing with who they worship or how they feel about Corellon personally. Rather, it seems to be tied to something innate, specifically relating to their souls.
Therefore, non-drow elven warlocks can trance just as any other elf does
An elf's class has nothing to do with their ability to trance, or what they see during it. Nor does their religion, or what god or being they follow. Rather, the nature of an elf's trance seems to be tied to their soul and their ancestry; put simply, whether or not they're drow.
The primal elves that became mortal and followed Lolth in her betrayal against Corellon were exiled to the Underdark, and eventually became drow. In this way, the nature of their trance seemingly has nothing to do with their current actions as mortals, and everything to do with their primal souls' past action of following Lolth. The only thing that might potentially contradict this is in the section on Arvandor (p. 48, emphasis mine):
Elves who live on Arvandor are no different from elves living anywhere else, except for the intensity of their passion. All manner of elves can be found there, including eladrin and even a few extraordinary drow. The splendor of the Seldarine illuminates their days, and their trances are filled with the intoxicating, blissful feeling engendered by their nearness to Corellon’s magnificence.
The section does not clarify how this interacts with the typical nature of a drow's trance.
Beyond this, the DM can always rule that a non-drow's trance is different in some way, depending on who they worship. Perhaps (for instance) an elf warlock who makes a contract with some evil being such as a demon or devil in exchange for their soul has a different sort of trance - like a drow's, or perhaps something entirely different. There are no rules for this sort of thing.
