As I was looking for ways to achieve this, I noticed you can change
the update mode of the RAID-1 to be "On Request". This could
effectively be used as backup, or I think it could.
In some way, yes. But it would be the exact same as leaving it to synchronize both drives automatically. Like some continuous data backup. However RAID solutions are about fault tolerance, not data backup. So you should never look at your RAID in the context of a backup. It's not a "backup of sorts", not even "almost a backup". It's not a backup. :)
For a discussion on this matter visit this question on ServerFault. But for a quick rundown, remember this: Does RAID allow you to recover data you just corrupted or accidentally deleted? Does a backup?
I have 2 Seagate Barracuda Green 2TB HDD in RAID 1 using Intel Rapid
Storage Technology. I use them as a storage volume, that is not
accessed very frequently, and most importantly, that is not written to
very often. This made me think that a continuous RAID-1 might not be
the best solution to ensuring the safety of my data as it is at risk
of accidental overwrite, viruses, and corrupted data.
Most definitely. And hence why I left this to last (the above would help put things in context).
What I would suggest perhaps is for you to drop the RST altogether from those two drives and use one to backup the other until you get an external hard drive. Once you do, you can put them in RST mode again and use the external drive as the drive to hold your backups.
Why an external hard drive? Ideally the target drive for your backups shouldn't be connected to the same PSU (or even wall socket) your hard drives are. Any power issues could burn both production and backup drives (and the PSU in your computer can do that). But an "external hard drive" can also be a remote location on the network, like the Cloud or some online backup service.
In any case, your priority at this point is not Fault Tolerance. As you are realizing, Fault Tolerance does not make sense until you resolve Data Backup. For this reason — and because you have an available drive doing fault tolerance — removing RST and do data backup instead, seems the sensible thing to do.