Everything in the original film trilogy either implicitly or explicitly points to the Rebellion knowing in some way Luke had some connection to “The Force,” but that did not change their strategies and methods. Remember, Jedi are there to help others in their battles but ultimately the non-Jedi members of the Rebellion would still have to fight their own battles knowing that—in many cases—the Jedi would support them in some, non-determinable way.
Also, Han bluntly states Luke is “crazy” in Return of the Jedi which might be connected to Luke’s “hokey” Jedi beliefs. And I think—considering how well regarded Han was with the Rebellion—that points to a simple scenario: Luke’s Jedi beliefs and ramblings about “The Force” are tolerated by the Rebellion since Luke has proven himself from day one to be a great pilot and a valuable fighter. Why mock the personal beliefs—and potentially alienate—someone who not only fights for the same freedom you fight for, but who also does an amazing job as a work-a-day fighter for your cause?
Star Wars (1977)
Obi-Wan was the last known Jedi and if he felt safe sharing his knowledge/identity with Luke at that point, it means that being a Jedi was something to hide from potential enemies; not allies or random folks.
Luke is introduced to Ben (Obi-Wan) Kenobi via a holographic message R2-D2 is carrying. Leia—a Rebellion leader—clearly knows who the Jedi are who who Obi-Wan is. R2-D2’s secrecy in who should see the message was simply a droid doing his job; this fancy lady told me that this guy should get this message and I don’t know who you are so buzz off.
So this random farm boy getting mixed into this mess was a happy coincidence; not only does Luke bring the message R2-D2 is carrying to Obi-Wan, after the message is delivered Obi-Wan gives Luke’s father’s lightsaber to Luke. At no point does he say, “Between me and you kid, these Rebels think we’re fruitcakes so be quiet.” Far from it.
The walrus man’s arm being severed in the cantina as well as Luke doing light-weight Jedi training on the Millennium Falcon were quite conspicuous. If being a Jedi was a “secret” demonstrating these powers in front of random drunks in a bar and in front of some random pirate at that same bar would not be a good way of keeping that secret. So one can infer that the Rebellion—who rewarded him a medal at the end of the film—would not only know about his Jedi skills but would be respectful of them as well.
The Empire Strikes Back (1980)
In the beginning of the film, Luke is carrying his lightsaber like any other weapon on his belt when doing a basic patrol mission. It’s not hidden away as some secret talisman. The depth of his Jedi training at this point was not that deep at all, but he wasn’t hiding it from his fellow Rebels. And even Han Solo knew he had a lightsaber on him when he had to cut open the Tauntaun to keep Luke warm.
Yes, his trip to Dagobah was done on the sly and without the knowledge of the rest of the Rebellion. But while not explicitly stated, it seems to me Luke was taking advantage of a lull in the battle—and the chaos of the overall struggle—to deal with something he personally found more important. Remember, he only decided to leave Dagobah when he sensed his friends were in trouble, but not because he didn’t check in with Rebel command.
Which is all to say—barring the events in Cloud City—Luke might have been AWOL (“Away Without Leave”) but he trusted his friends to be able to regroup without him and he would find them again when the time came. And the Rebellion was such a rag-tag endeavor, they seemed to be understanding of the idea that their volunteer fighters would rejoin them when they could… Not always when they are needed.
Return of the Jedi (1983)
I believe that it’s strongly established right at the beginning of Return of the Jedi that Luke is very public and vocal about him being a Jedi Knight. For example, when R2-D2 delivered the message from Luke to Jabba the Hutt on Tatooine, Luke very clearly introduces himself as a Jedi Knight; bold emphasis is mine:
Greetings, Exalted One. Allow me to introduce myself. I am Luke
Skywalker, Jedi Knight and friend to Captain Solo.
Perhaps the idea that C-3PO was stunned when he and R2-D2 were given to Jabba as gifts implies Luke was hiding the true contents of the message. But I believe the in-universe logic has been that C-3PO is a blabbermouth, but R2-D2 can keep a secret. So everyone else in on this rescue knows who/what Luke is… Except for C-3PO who they could have fed some nonsense to in an effort to distract him from knowing the truth.
Regardless, at this point not only is Luke clearly identifying himself as a Jedi Knight, but he’s also not afraid to share that fact with Jabba the Hutt and everyone else in his palace. He has no fear of being exposed to jerks like that in any way, so I can’t imagine he would be hiding that fact from the leaders and ground troops in the Rebellion.
Further, when Han is freed from the carbonite but then locked up with Chewbacca in Jabba’s palace this exchange happens; again bold emphasis is mine and video of the dialogue below as well:
[Chewie barks an excited blue streak.]
HAN: Luke? Luke’s crazy. He can’t even take care of himself, much less rescue anybody.
[Chewie barks a reply.]
HAN: A… Jedi Knight? I—I’m out of it for a little while, everybody gets delusions of grandeur.
[Chewie growls insistently. He holds Han to his chest and pets his
HAN: I’m all right, pal. I’m all right.
Of course Han is oblivious since he’s been out of it since the events of The Empire Strikes Back. But again, Chewbacca is no blabber-mouth and would’t jeopardize their rescue mission—even though it seemed to be going badly at this point—just to gossip with Han.
But the major point being: Luke clearly identified himself as a Jedi Knight to those he trusted. I can’t imagine why he would hide this knowledge from the leaders of the Rebellion. And I can’t imagine how he might have been used by the Rebellion to further their cause past what we know he wants to do: Confront Darth Vader.
People seem to respect Luke’s beliefs/powers, and have seen—or at least have heard—about what he can do but there is a lot of mutual respect here: The Rebels just want to defeat the Empire and Luke—as a Rebel—wants to defeat the Empire as well. But Luke strongly believes confronting his father on his own is part of the way this end-goal can be achieved but he has no idea how any of that will play out; he just “knows” it. That said, Luke’s spiritual quest for closure won’t stand in the way of the Rebels militaristic approach and the Rebels won’t stand in the way of Luke’s personal quest.
Past any of that, Luke ultimately felt quite uncomfortable on non-Jedi related Rebel missions such as when they are flying the stolen Imperial shuttle to Endor to deactivate the shield:
LUKE: I’m endangering the mission. I shouldn’t have come.
He was clearly conflicted about being on that mission to Endor, but it all worked out in the end.