Want to minimise cost? 128 GB USB drives can be had for around ten dollars. Fifteen dollars will get you 256 GB. Stick with a cheap but reliable brand like SanDisk or PNY. It will last you years with proper care, even with regular use.
I do not really see a reason to invest in a "real" hard drive or external SSD for your use case. There isn't enough data to justify it and USB drives can take a lot of abuse and still work.
Edit: There are a lot of fake USB drives sold online where the controller has been hacked to report a higher capacity to your operating system than actually exists, for example, reporting 512 GB when there's actually only 64 GB of storage. If you try to store more than the actual capacity, your old files will be overwritten with the new ones. That's why you should be suspicious of very high capacity drives (1 TB+) sold online for low prices. I would not buy any USB drive online that claims to have a capacity greater than 512 GB.
Regarding fakes, there are tools you can (and should) run on any new drive. I personally like h2testw, but there are others as well.
You should also avoid sketchy resellers. This includes Amazon, due to their policy of commingling and their response to counterfeits. Don't buy from Amazon! Stick to trustworthy and first-party sellers. If buying online, make sure they don't have a third party seller. In fact, probably best to skip any that even have a 3rd party "marketplace" system.
I don't think there are really any fake 256 GB drives on the market. The real drives are priced low enough that there's really not much profit to be made from selling fakes. It's just not that much cheaper to make a 32 GB drive and flash fake firmware than to actually make a legitimate 256 GB drive. Or buy the AmazonBasics brand since only Amazon makes and sells them.