To elaborate on this, make sure you're visiting the lemmy.world community from your home instance, not going to lemmy.world's URL. Use the search button (upper right corner) to search for the community, and the result will be the link to that community from your instance.
Yeah, this is a downside of federation right now. Too many linking still goes to the target instance instead of loading it in your home instance. I hear Mastodon is fixing this behavior, Lemmy should definitely follow this, imho. (Making hxxps://instance.tld/@\user equal to @user@\instance.tld)
Your login is unique to the home instance. In effect your username is the full vestmoria@linux.community but the @ portion gets added in when you log in on the linux.community page. There is some theoretical talk of federated identity that would let you use credentials across instances but the logistics of it seems daunting.
You can still read and post to wherever from your home instance, so long as the other server wasn't defederated for some reason.