So I'm wondering if there is an NVR that has motion sensing built into it that would then alert me somehow there is movement etc.
The newer NVR's today have both video motion algorithm and analog sensor terminals built in to the NVR.
I utilize a software NVR called ZoneMinder (http://zoneminder.com) which runs in Linux.
Software like Blue Iris utilizes pure video motion algorithms for triggering recording events.
Most if not all of the small doorbell cameras have a motion detector / camera built in and proprietary software which talks to the cloud and a proprietary smart phone app / internet app for video events.
Over 50% of the package delivery folks today never ring the doorbell anyhow.
Not sure if the small combo doorbells trigger a doorbell ring when not pressing on the doorbell button.
Many years ago my solution here was to utilize a combo camera with built in analog sensors. (IE: PIR, night and day and intrusion). I DIY'd the whole thing using ZM and my alarm panel (OP2). Long math way here split the video into digital and analog components. My combo camera/pir's do not resemble cameras at all which I like as the new ones all do look like cameras.
Works fine today.
You can purchase a combination outdoor camera/pir today except that it is bigger than a doorbell button.
The doorbell ring (with Elk combo 960/930) provided one trigger of many. I used PIRs (and walk sensors) and driveway sensors to provide a second trigger and door bell ring. The events are software driven with stills and motion. A package delivery video includes walking from the truck to the door.
Personally here would like to find a small footprint doorbell button with HD and PIR for a DIY solution.
Simplest is just to purchase a door bell camera and utilize the door bell camera propitiatory software. Integrating something like this to an NVR is going to be a bit difficult unless the mfg of the camera provides an API for tapping into their application.
Many of the long time CT members here were doing this before the small combo doorbell button came to being using an analogue solution and getting triggers to play back on their phones using software solutions.
As mentioned above it is what you want to do.