Apps demand I "Update Google Play Services" on MIUI - but should I?
Yes, that message is valid as your Google Play Services (GPS) version is relatively out of date (current version is 12.2.17). If you need to use the applications that require you to update GPS, then your only option to remove that message is to update.

The easiest way to update your GPS is to follow the notification. If for some reason that doesn't work for you, APKMirror has all the GPS variant here. The APKMirror route does require you to know a fair bit about your phone though to find the correct variant.