I have a server running on an ext4 fs. It has a HD with a btrfs-formatted partition, /laptop. /laptop is exported r/w to the local network.

I have a laptop (surprise) which is *all* btrfs and mounts the exported filesystem as /backup. So far, so good.

Since both partitions (/ and /backup) are btrfs, I was hoping it would be possible to easily do backups with btrfs' superb snapshotting capabilities. But when trying to send/receive, I get the following error:

linux:/ btrfs send /home/BACKUP | btrfs receive /backup/home
At subvol /home/BACKUP
ERROR: /backup/home doesn't belong to btrfs mount point
ERROR: failed to dump stream: Broken pipe

So I think maybe I have to first create the subvolume on the server:

linux:/ # btrfs subvolume create /backup/home
ERROR: not a btrfs filesystem: /backup

Being relatively new to both NFS and btrfs, I'm a bit overwhelmed by the intricacies of them working (or not) together. Any hints as to what I'm doing wrong?

You may (still) need to do the receive "at" the server. The scripts pointed to by the wiki use ssh to connect to the server and do the receive.
Once that is all working, you can use incremental send/receive.

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Use ssh to access a mounted filesystem? Isn't that redundant? Maybe you meant to tell me that NFS isn't any use in this context.