How to add new persistent disk without rebooting the server?

It's actually very simple.
Your screenshot shows that your disk-2 is not attached to your VM instance.
It should look like this

And now compare to what you posted.

So what you need to do is this:

1. Head to "VM Instances"
2. Click on your Instance
3. In the top click on Edit

4. Scroll down to "Additional disks" section and click Add

5. Select your disk from the drop down menu

6. Scroll down to the bottom and click Save.

It should now show up as sdc in the dmesg output

or in the fdisk -l

After some testing, I believe your issue is related to the use of a blank disk. In my case, I attached to the VM one blank disk and one formatted disk. The formatted disk got recognized as ¨/dev/sdc1¨ but the blank disk was not showing.

i.e

ls -altr /dev/disk/by-uuid/
total 0
lrwxrwxrwx 1 root root  10 Nov 11 16:50 40206d63-43a8-4d1c-8588-581cefb022a3 -> ../../sda1
lrwxrwxrwx 1 root root  10 Nov 11 17:34 7f63bd57-6ccd-4cea-b0f0-9544870bf67a -> ../../sdc1

¨fdisk -l¨ showed the disk was not partitioned but confirmed it was attached to the VM as /dev/sdb

sudo fdisk -l
Disk /dev/sda: 10 GiB, 10737418240 bytes, 20971520 sectors
.....
/dev/sda1  *     4096 20971519 20967424  10G 83 Linux
Disk /dev/sdb: 10 GiB, 10737418240 bytes, 20971520 sectors
Units: sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 4096 bytes
I/O size (minimum/optimal): 4096 bytes / 4096 bytes
Disk /dev/sdc: 10 GiB, 10737418240 bytes, 20971520 sectors
.....
/dev/sdc1  *     2048 20971519 20969472  10G 83 Linux

I used ¨fdisk /dev/sdb¨ to create a partition table, then ¨fdisk -l¨ displayed

Disk /dev/sdb: 10 GiB, 10737418240 bytes, 20971520 sectors
Units: sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 4096 bytes
I/O size (minimum/optimal): 4096 bytes / 4096 bytes
Disklabel type: dos
Disk identifier: 0x5808f242

Device     Boot Start      End  Sectors Size Id Type
/dev/sdb1        2048 20971519 20969472  10G 83 Linux

As per the document I formatted the disk

sudo mkfs.ext4 -F -E lazy_itable_init=0,lazy_journal_init=0,discard /dev/sdb1

Finally the disk showed

ls -altr /dev/disk/by-uuid/
total 0
lrwxrwxrwx 1 root root  10 Nov 11 16:50 40206d63-43a8-4d1c-8588-581cefb022a3 -> ../../sda1
lrwxrwxrwx 1 root root  10 Nov 11 17:34 7f63bd57-6ccd-4cea-b0f0-9544870bf67a -> ../../sdc1
lrwxrwxrwx 1 root root   9 Nov 11 18:09 460a2e04-0f66-42d4-b9cc-8567aa6ee7b8 -> ../../sdb1