i think your requirements are very demanding and sometimes contraddicting
so first of all forget about "under 100$" if you want more than just a crappy, unsecure, not really waterproof ip camera permanently connected to the external world
1080p will need a LOT of bandwidth, a fast router and an excellent wifi link
will require lots of disk space for storing video/snapshots.
wifi transmission needs a lot of power and that will require a large solar panel/battery and electronics to handle that. not sure solar powered camera exists, i have never seen one.
most cameras are inherently not secure as video transmission protocols are not encrypted
i have a few ip cameras and they all come with very questionable default setting regarding upnp, ddns, cloud storage and access, and sometimes is very hard or impossible to disable these features.
in the past ip cameras have been used as nodes for DDOS attacks so looks like many chinese cameras (arent all of them chinese nowadays) have backdoors or very bad security.
cloud cameras are, from this perspective, even less secure
what i would do is the following:
get an IP camera that more or less satisfies your requirements, set it up at a reasonable resolution that will still allow video transmission at an acceptable frame rate and that will not kill your wifi network. for surveiilance purpose you dont need HD
set up a rpi with a usb disk to store video stream, you can find online a lot of info on how to do it. this will have to be inside your home network, so is secure and is a very cheap solution.
set up door knock on your router, so the camera port forwarding will not be active all the time (believe me, you dont want ports always forwarded to the external world) and noboody will be able to access the camera from the ext internet. not sure there are routers with this feature as standard, i use custom router firmware (dd-wrt)
to do all this you need knowledge on networking, router firmware, linux, and how to set up a rpi