There are three stages of integration into Atlantis that are easily visible on Earth. There are, of course, infinite gradations visible in Atlantis, from the "holy crap, Atlantis!" stage to the "I just got turned into a woman" shrug and carry on like nothing has changed stage, but those aren't visible on Earth.
Earth, of course, only sees Lanteans rarely, even with the gate bridge, because Atlantis is small enough that it struggles to compensate for even one person going to Earth even if they're not somebody who's key to the day-to-day operations of the city. So the three stages are fairly concrete.
The first stage is seen mostly on those who decide to return to Earth after only a small amount of time, after finding that they can't cope with the myriad troubles the Pegasus Galaxy heaps on even the most lab-bound scientist, much less those on gate teams. It's most visible in how they surrender their weapons quickly, sometimes with relief. To them, surrendering the weapons that nearly everybody in Atlantis carries as a matter of habit is a sign that they're safe. It's Earth, what could they possibly need weapons for here?
The second group comes more rarely. This is the group that has started to identify with Atlantis. They're not leaving on a permanent basis unless they're too injured to go back, but they have people on Earth that they want to visit and they're still willing to take vacations on Earth. They surrender their weapons a little reluctantly, because Earth isn't quite home for them anymore. It's all slightly alien to them, and that makes it feel like an away mission. But they're not quite at the stage that they really see it as an away mission.
The last group never sets foot on Earth if they can help it. Earth means giving up weapons they never allow more than two feet from them even in sleep. It's not going home or a vacation to go to Earth, it's an offworld mission with not only the risk of never being able to go home to Atlantis (because that's always a risk on offworld missions) but also never being able to be who they are, and of getting the expedition cancelled completely. To them, going to Earth is a terror-wracked torture session, no matter how long they're there for. Earth foods taste strange to their palates. Earth beds are too wide. There's no sound of the ocean gently lapping at the piers to lull them to sleep. Most of the original expedition members, who'd halfway expected to never make contact with Earth again, easily fit into this group. Atlantis is their home, even if they were born on Earth.
When Atlantis declares independence, it's only the first group that returns to Earth.
