Anabolitika?

Pilsator

Network "sign-in" webpage?

DNS-Names are not bound/specific to a protocol/service type. Of course some might fit better for a serivce, e.g. the top level domain ".im" fits "Instant Messaging" (e.g xmpp, matrix ...) services well. But still you could host a Website (https-protocol) on the same address. The list of valid top level domains is not static. new TLDs are being approved constantly. This means e.g. ".world" is a valid Top Level Domain but not many/big services use it for now, so it is rather unknown.

Officially it is the domain of "Mali" https://en.m.wikipedia.org/wiki/.ml Might be also the letters "l" and "m" which are the consonants of lemmy There is no ".lm" top level domain

It would replicate this one specific group/chat only - from the moment of subscription. Depending on the load it can get heavy on your instance.

Matrix has one caveat: it synchronizes every room (group chat) from another instance to your instance fully- which one user subscribed to on your instance. Because of this the instance-systems/servers are under heavy load for private userage (not controllable number of users and chats). Many governmental institutions (controllable number of users and chats) use though, because in case of "disasters/incidents" the data is not lost but saved all over all replicas.

Use a dedicated DNS-service on your local network which has the local IPs in it's DB. Use that DNS-service as your first/primary DNS on your local network (settings).

Could be, that they are out of sync. Maybe overloaded. Not enough capacity/bandwidth?

Hey, why you think it gets overloaded? Is something slower / lagging in synchronization?