Hi Peter,
Only just now seeing this. Hopefully it can be of some help.
I've had some pretty close calls losing data in the past, so I'm pretty paranoid about backup and data security. I was a network administrator for an entire ad agency for many years, in charge of insuring that no one lost any of the important client files they were working on. I come by data backup paranoia pretty honestly I think. Those days are long past, but precious family photos and my PhD dissertation meant that I had to think seriously about how I would secure my own data...without the resources of an ad agency behind me.
A few important points you should know:
1. As you've seen, mechanical hard drives die and, contrary to what some people might think, SSDs die too. In fact, when an SSD dies in can be a lot worse than a mechanical HD because recovery because the data is more difficult to recover.
2. A drive (HD or SSD) connected to your laptop backing up your laptop files is great, but this is not a complete solution. What happens if both the laptop drive and the backup drive die? Answer: you're screwed. Solution: RAID. Your backup configuration should be two drives (HD or SSD) that are mirrored. They both have to be the same type and size. For example, two 1TB SSDs. If one drive dies, it can easily be rebuilt with the mirrored data on the other RAID drive. BTW, RAID stands for Redundant Array of Inexpensive Disks. There are plenty of two drive RAID backup systems on the market. Drive backups should happen at least weekly. Nightly is better.
3. OK, now you have a RAID backup drive, what happens if your apartment/house burns down? Again, you're screwed. This happened to a colleague of mine. His apartment burned to the ground and all of his dissertation and its associated data went up with it. Pretty disastrous. This made me doubly paranoid during my PhD program. Solution: off-site storage. This can take two forms: cloud storage or physical offsite. I use a cloud storage service called Dropbox, though there are tons of others that work just as well. All of the key files I consider super important are synced to the cloud service. Whenever I make a change locally, the change is replicated on the cloud storage service immediately. All syncing is automatic. Physical off-site storage is the practice of physically taking a duplicate drive of data and storing it in another location. It's effectively the same result as cloud storage, just slower, less frequent, and a lot more annoying. Cloud storage is the way to go.
4. Having two laptops is a fine solution. It's actually what I do. I have a laptop at work and one at home. The data is always synced between them using the cloud service. The cloud service is crucial in a two computer setup. Strictly speaking, I have three computers that share data across my cloud services, my work laptop, my home laptop, and my home music studio computer. DO NOT do what my other colleague does and keep files stored on a USB thumb drive and plug it in to whatever computer you need to work on. This is a profoundly bad way of managing data. Those drives die, you can lose them, accidentally leave them at home when you need them, and so on. The best way to handle data sharing across devices is to use a cloud data sync service and select which files you want each computer to sync. For example, I don't need any of my academic work on my music computer, but I do want to backup my musical creations. So I tell dropbox to backup my music folder and also tell it that I don't need a copy of my academic work on the music computer. By default, Dropbox wants to share all data with all computers in your setup, but you don't have to do it that way if you don't want to.
I have drawn up a quick diagram that hopefully explains it. I'm a visual person, so these help me.
Is this all overkill? No. Once, you have it all up and running—which really isn't hard—the peace-of-mind is priceless. I never worry about data anymore. If I woke up one morning and one of my computer blew up overnight, I'd just go out and buy and new one and be up and running in a few hours. Installing my apps would be the longest part of the process.
Let me know if you have any questions. This stuff—along with web site and application design—used to be my bread-and-butter before I climbed the Ivory Tower.

Brian
PS: I should mention, my ACTUAL network backup system is more complex than this. I actually use a NAS (network attached storage) system instead of a physically connected backup drive. It's essentially a RIAD that sits on the network that can back up any computer on our home network. And there are lots of them, seven at last count...yikes