I expect something like a push notification saying: "Tom entred your property" or "unknown person entred your property" based on the face and this line.
That combination is sort of science fiction, if you expect it to work regularly. To the extent that face recognition works, it works in tighter / closer range while typically line crossing is used in wider / farther out areas.
Also, facial recognition built into cameras (i.e., the camera determining it's john or jay or joseph, etc.) is quite rare while facial detection built into cameras (i.e., the camera determines there is a face in the picture but without determining who's face) is more common.
All that noted, $500 in 2017 for an IP camera can get you quite a lot. When you say "Secure" software (not like HIK or Dahua)" do you mean no Hikvision or Dahua cameras allowed or just that you don't want to use their VMS / NVRs? If you are open to using Hik or Dahua cameras, there are clearly lots of cameras that meet your specifications.
Beyond that, the two most common higher-end cameras chosen by integrators are Axis and Avigilon, though Bosch, Hanwha/Samsung, Panasonic, Sony, to name a few are found in this category.
Take a look at our Camera Finder where you can enter in what specs you want to pick cameras. It will return a list of matching cameras including within a price range, e.g., here are 30 cameras, 1080+, true WDR, IR >20m, bullet form factor.
Btw, you will have a lot more options if you are open to domes as advanced features are more common in domes than bullets.