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I just confirmed it. Yes, the visualization of a profile (the fediverse community located in lemmy.world in this case), will vary depending on the instance being used to access it. It’s really a pity though. Things like that should be transparent.
IMHO, whenever the user searches explicitly for a profile/toot/post/whatever, all relevant data should be fetched as well.
Although it’s possible to follow Lemmy’s communities in Mastodon, I would not suggest it as the experience is not the same. That’s my reason to have separate accounts.
If you only want to be “notified” of new posts in a given community, I think you can subscribe to some sort of RSS feed. In Calckey (now Firefish), you can add timelines that will show specific RSSs.