The pragmatic answer
Being pragmatic, video game genres are labels (tags) that people use to find said video games.
These labels or tags don't need to be mutually exclusive. And there might be synonyms. That last part should be obvious once you consider the range of language and cultures of the world.
People are going to find your game by the genre labels you pick on whatever distribution platform you choose. And thus, you should pick labels that have games that are similar to it.
This also suggest that the criteria for what makes a game of an specific genre is constantly evolving.
So, chiefly, the genre is about player expectations. People might feel betrayed if they purchase something labelled "Immersive Sim" but don't get something that resembles their idea of "Immersive Sim".
It also also about skill transference. Games of a genre build on top of an existing library, and expect some basic familiarity with the mechanics※, which they can use as affordances in their game design.
※: It is very rare that a game targets absolute beginners, and teach the mechanics from the ground up. This is, in part, because the target audience that already knows what they want, are willing to pay for it, and are vocal about it… Are not the beginners.
Thus you can answer you question by looking at other games in the genre. This should not be a determinant for your game design, but you might want to be careful when picking the genre labels.
But can you find an "Immersive Sim" label on your intended distribution platform? Chances are you can't. Many people have played "Immersive Sim" without knowing the term. And thus, from a pragmatic point of view, the exercise might be futile.
What you could do, is look for games that you recognize as "Immersive Sim" and see which of the genre labels the have apply to you.
Make your game, the way you want, and worry about it later.
The not-pragmatic answer
Video game genres are an attempt to capture and communicate the reality of the games. There aren't perfect examples of video game genres. Instead they are all approximations to a platonic ideal. And the reason why the genre seems to evolve is because we learn more about said platonic ideal through the exploration of the approximations… Which is also limited and influenced by technology.
Since we cannot study the platonic ideal directly, we have to resource to find the commonality between the approximations and codify it.
The position that Immersive Sims don't have to be first person makes sense if we consider the perspective an artifact of technological limitations. Even thought the same argument would suggest that the genre should move from first person to virtual reality… It does not preclude making third person Immersive Sims.
The reason we have the argument is because the genre was - arguably (we'll get to that) - born in the first person perspective. Since the - arguably - first game of the genre is Ultima Underworld, published by Origin Systems and produced by Warren Spector. It is important to notice that most of the games in the Immerisve Sim genre trace back to Warren Spector. Without diminishing the influence of other important designers in the Immersive Sim genre such as Paul Neurath, Harvey Smith (Deus Ex) and Ken Levine (Bioshock).
Ultima Underworld was also a first person game before first person shooters were a thing. And most of Immersive Sims remained first person: System Shock, Thief, Deus Ex… Prey… BioShock… Dishonored…
Looking at the games in the genre I find these commonalities: first person, single player, narrative driven, RPG elements, and emergent game play. It is worth noting that the genre does not require open worlds.
However, a relatively early notable game that is considered an Immersive Sim which bends the rule is Vampire: The Masquerade – Bloodlines, which has both first and third person perspectives. The game is also notable that it was outsider… I mean, it cannot be traced back to Warren Spector.
More recently we have Weird West, which is arguably an hybrid of Immersive Sim and Top-Down shooter, or a Top-Down shooter with "Immersive Sim elements", yet it is also notable because we can argue it is in the Immersive Sim legacy: Weird West was developed by Raphaël Colantonio's WolfEye Studios, Raphaël Colantonio previously was the founder of Arkane Studios, and before that he was part of Electronic Arts working in the french localization of Origin System games - including System Shock, of which Warren Spector was producer.
We can also look at the precursors. Ultima Underworld could be described as a first person dungeon crawler RPG, except not everything is out to kill you. As such, it comes from an even longer line of games… And when do they approach the Immersive Sim platonic ideal to merit the name is unclear.
In fact, Raphaël Colantonio had argued and defends the position that Immersive Sims can be top down. And considers the genre can be traced back to Ultima VI: The False Prophet (source)… A point that came up - in part - because Wikipedia currently says on Immersive Sim:
"Warren Spector considered Ultima VI: The False Prophet (1990) the first game to have an immersive sim mentality as while played from a top-down view, it relied less on events and planned-out puzzles, and instead provided the rulesets and systems through its living world to allow players to craft their own solutions to situations."
Which is not sourced. But probably is based on the article How Warren Spector Created a Genre, and Set Games Free, which says "Ultima VI: The False Prophet, which [Warren] Spector considers to be the first immersive sim", which I suspect to be a misquote… But if it ins't then probably we should not be counting Ultima Underworld as the first in the genre, but Ultima VI: The False Prophet, which had a top-down-esque oblique perspective.
Heck, we should not have Warren Spector as gatekeeper of what an Immersive Sim is anyway.
Thus, yes, you can make an Immersive Sim with a third person perspective. Can you still call it Immersive Sim? Sure, nobody is stopping you. Is it correct to call it so? Arguably. But it is certainly not a traditional Immersive Sim. You could go by saying "Immersive Sim elements" if you find it too controversial.