The short answer is yes, the Z2 has its own 'pedometer'. By that I mean that the device captures 'steps' by itself. You can install a third party app (there are loads on the Play Store - here is an example
) to view the number of steps taken. The sensor actually counts the steps constantly, the apps just 'plug into' this data and present it to you.
'Moves' as you are aware does this too, but it also tracks your location. it is in my experience too battery hungry to be worth the installation.
The 'Lifelog' app along with the smartband is quite useful, but it is the smartband which counts the steps taken. The app shows more than just steps, it shows time spent browsing etc.
The other issue is the difference between counting steps via the phone and via a third party piece of hardware (smartband etc). If you have your phone with you always, then it's not too bad, but if like me, you leave your phone for periods (when I go out for lunch at work), then it won't be counting your steps whilst sat on your desk.
My own preferred choice is a Fitbit device, which I can use via all of my devices (PC, tablet, windows phone etc). But it is all a matter of personal preference.