…cracks knuckles…
There are several companies I would take a look at… All of these fit your requirements, the button you are asking about is usually referred to as a “panic button” and you can (should?) stash several of them around the house in concealed but accessible locations. They also offer alarms for stuff like your water heater floods the basement.
All of these are DIY, meaning you buy the kit and install it yourself. You can buy as small or as big a kit as you need, motion sensors or not, video cameras or not, etc, etc. All of them have some sort of monthly subscription, but you can cancel anytime and you aren’t out a fortune like some other companies (cough ADT cough).
Simplisafe - started as an alarm company, has expanded into video, doorbells, etc. Doesn’t work without subscription, subscription includes professional monitoring which calls 911 for you if needed.
Ring - Started as a doorbell video system, has expanded into video and alarms. Is owned by amazon. Doesn’t work without subscription, subscription includes professional monitoring which calls 911 for you if needed.
Abode - More similar to Simplisafe (alarm first). Is the only one capable of 100% DIY. Has a full professional monitoring subscription available as well, but also has a cheaper plan that sends you the notifications and you decide whether to call 911 or not.
CAMERAS
From a privacy perspective, think very carefully about where you might put a camera that sends 100% of the video out to the internet where some 3rd party has access to… all of it. For things like an exterior doorbell camera, I don’t have that much of a concern (its likely public), but for interior cameras, I am not a fan.
From a usefulness perspective, be aware that a camera pointing at your living room with a TV in it will have a ton of false positives and will it catch someone who breaks in through the kitchen window? So ideally they should be aimed at places where an intruder is likely to walk through or at things of particularly high value. Some cameras are easier hidden than others.
If its in the budget, and you have the expertise, I’d recommend keeping your internal cameras private and stored only to places/devices you control with something like BlueIris. This software can control up to 64 cameras (I think) and you control where you do or don’t want to send video for storage. Then just have the external cameras controlled by whichever alarm system provider you choose.
CONNECTION
Most of these will connect to your home wifi/internet. But you also NEED a secondary connection, usually this is a cell provider (I believe all 3 include this as part of their subscription) in the event someone cuts your cables or the power just plain goes out. This is pretty standard nowadays, but make sure whoever you choose has this option.