As the title states, I currently have my Synology NAS with a single 4TB drive in it. I now wish to expand this with another 4TB drive. If I expand the volume (it's currently setup as SHR) will the initial disk be erased? I know it says "existing data will be erased" when you select the new drive, but I'm not sure if it means only on the new drive or both the new and the existing drive.
No the existing data will not be erased when you add a disk to the SHR array. The message is simply saying the new drive being added will be formatted.
You data and applications will actually remain available during the expansion process; there is no need at all to even stop using the NAS whilst SHR Vertical Expansion is taking place.
That said - before any type of RAID manipulation of any flavor you should backup first.
Finally it is worth pointing out that if you are using two equally sized disks there is no point at all using SHR over RAID1 - indeed you should use RAID1 as SHR will incur an overhead in both read and write speeds compared with RAID1. The advantage of SHR comes when using odd sized disks.
I cannot say specifically, however - if you add a disk to a RAID config, it'll need to be wiped to become part of your RAID. Losing that data is - as you surmised - inevitable.
When it comes to the original data though, the answer is a little more complex - RAID, depending on type, will differ in where it writes data in a two drive configuration, as opposed to a one drive config. (Aside from straight mirroring). This means your array needs to re-layout the data on the disk; the simplest way to do this is reformatting, wipe everything and start again.
That's not particularly useful for most people, so a lot of RAID devices do on the fly rebuilding - they effectively have to when you replace a drive anyway.
Referring to: http://www.synology.com/en-uk/support/tutorials/559 I note that it says: For SHR volumes, storage capacity is only expanded when adding hard disks to a volume with two or more disks.
So you may simply be adding redundancy by adding a drive. A spot of googling suggests this to be true - 2 disk SHR is basically RAID1 - mirrored data, no capacity. As such, it's extremely unlikely to be losing any of your source data. (But won't be giving your more capacity either).
If you want to increase capacity, you'll need to switch to RAID0. However this has a very significant caveat - RAID0 does not protect your data. It is WORSE than a single drive for reliability, because if either drive fails, you lose data. RAID0 is generally a bad plan for anything you care about keeping. (And from a brief bit of digging, swapping RAID types is destructive).