Hosting
Your data is reachable only while a store holding it is online.
Local First
Edits can be made offline and synced on reconnect. The local instance also caches external data you fetch, keeping a copy on disk for a time. Peers sync directly with each other when both are online; a hosted store is not a middleman for this, just another peer that happens to stay awake.
Availability
Asking a store to host a document is how it stays available while your devices are asleep. A host replicates the documents it agreed to host, pins their content so it is not garbage collected, and counts the bytes against your quota — without interpreting what it stores. It sees sizes and hashes, never meaning, so there is no format it can reject and no schema it can force.
Hosts provide availability, not authority. Hosting also says nothing about who can find your data; that is discovery, a separate decision.