apk - How do I know which version of Google Play Services to update to?

I have an Android 5.1 system on a Doogee X5 phone, but this question is a bit more general.

Suppose I want to update my "Google Play Services" manually, without having a Google Account of any kind. This question tells me where to go to get the APK, but - how can I tell which APK I need? Wouldn't there be different APKs for different versions of Android, and possibly different vendors (who may provide modified Android versions)?