The first messages (before typing the password) are normal: the system is searching for the volume group, but cannot find it because it's on an encrypted disk/partition.
In other words, this message is normal and expected on current Debian 10 and can be ignored.
(If anyone knows how to tell Debian 10 initramfs that the LUKS encryption should be unlocked before making the first attempt to probe for LVM, that knowledge would be appreciated.)
The second set of messages suggests that the system is also trying to mount the filesystem on LV
vg/home before the encrypted disk is unlocked.
With the default configuration, the initramfs boot phase will only aim to unlock just the one encrypted volume that contains your root filesystem, and activate just the one LV that contains your root filesystem. Anything more than that is left for the "main" boot process, after the system has transitioned from initramfs to using the real root filesystem. If you need multiple LUKS encrypted volumes unlocked or multiple filesystems mounted in the initramfs phase, you will need to explicitly configure that.
Have you done something to cause the initramfs to mount your
/home filesystem? Unless you're doing something very special, that is probably not necessary. But even so, the system will make further attempts to activate any available LVs and mount all the filesystems listed in
/etc/fstab later in the boot process, so the second set of messages may also be ignorable.
You should probably find out which device the error message
Couldn't find device with uuid ***-***-***-*** refers to, by noting down the UUID and once the system is up and running, using
lsblk -o +UUID or
blkid to list the UUIDs of all the filesystems and identifying the matching device. If it is not
sdc2_crypt, part of your
/home filesystem might not be fully contained within the encrypted volume, which will cause some files on that filesystem to not be protected by the encryption.
lvdisplay --maps /dev/vg/home to see where all the parts of the
vg/home LV are located. If under
--- Segments --- it mentions any physical volumes that are not encrypted volumes (like
sdc2_crypt), you have this problem. If you have enough unallocated space on an encrypted volume that belongs to the same LVM volume group (check with the
pvs command), you can use
pvmove to move the wayward parts of the LV onto an encrypted PV - even while the
/home filesystem is mounted and in use.