I think the Lorex is exactly what you want, with remote app access or a VPN configuration, where the remote access is turned off, but you could VPN into your local network and use an iPhone as if it were on the LAN instead of via the Internet. That keeps all video local, no cloud traffic, etc.
You can have the Lorex push alerts, though that requires letting the company have some interaction with your system. I believe it’s all outward: your system pushes an alert to Lorex, which passes it through the app.
The local storage mitigates cloud storage, and you can get the minimum 1TB or 2TB drive and just store motion clips, or store full-res video with a larger drive for a long period of time. The Lorex systems all do video compression, H.264 typically, IIRC.