As of a recent update to iTunes for computer, the second option no longer works. There is no way to purchase apps on the aforementioned new iTunes. (If there is a way, I simply do not see it.) It was stated that the computer version is "shifting to a focus on music," and thus we are without a method of downloading older apps to legacy iOS versions without buying or owning an Apple device that is compatible with the up-to-date version of the app in question. If you, dear reader and/or author, know of a work-around to install older app versions that does not involve jail-breaking- please leave a reply.
Download itunes 12.6.3 from this link by clinking on the button that says 32-bit. This version of itunes still has the app store Ignore the fact it says this version is for enterprises. You can still download it anyways. https://support.apple.com/en-gb/HT208079
Sometimes you just don't want to buy the latest, coolest gadget; or upgrade to the latest OS. Sometimes you're very happy with what you have, and that's completely ok.
But then sometimes you get left behind and, although your gadget still works fine; and you're not using pastels on a daily basis, not all apps remain available for your trusty friend.
If you're using an older Apple device and can't, or just don't want to upgrade to the latest operating system, but you really love Spotify (of course you do, otherwise you would have stopped reading long before getting here (thank you, btw)), then you still have options.
Not too long ago Apple enabled everyone with an older device, running iOS 4.2 or later, to still get those apps they love; that includes you!
If you previously installed Spotify on one of your iDevices, you'll find it in your device's App Store under the Purchased tab.
If you didn't previously install, or you're using a new Apple account, just go to the App Store in iTunes and download Spotify. It doesn't matter what version that is; you just need to have Spotify as a "Purchased" app on your Apple account.
Now you can go back to the Purchased tab on your device, and you should find Spotify there. Hit that Install button and you should be offered the option to download a previous version.
Both options should work for everyone, but we've had reports of folks in some countries not being offered the previous version. If this happens to you, you can try making a US Apple account and following the same steps, above.