I have an issue with rainfall running into the back of my my property and into my house through patio doors.
Would a sump pump start at 2-3 inches of water stopping it from entering my home. When this happens, I usually wind up with 5-8 inches
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DISCLAIMER: Not explicitly an answer to your question and definitely not what you're going to want to hear. Also, not a landscape expert.
I believe that this is going to take some landscaping instead of or in addition to a pump. Sloping the land away from the house would be ideal. If your house is (unfortunately) on a low spot, you may need to put in a bit of a ditch across the property to catch the water and direct it away from the house. Note that this doesn't need to be an actual trench, just a gentle valley that's lower than the house's elevation - it can still be "flat" enough for the kids to play soccer on and barely notice the ball rolling on its own to the center of the lawn.
This valley would then run the water off your property to another natural low spot where it can continue naturally draining to the city's storm drains, local creek, or whatever. (Note that you'll not want to just dump it in the neighbor's yard unless you really don't like him and don't want to try to repair the relationship and are willing to open yourself to the possibility of lawsuits!) You may need to have a dry well or cistern dug to contain all the runoff. It may be possible to tie this directly into the city storm drains or at the least, run your sump pump from here to dump it to the city.
Depending on the soil type in your area, you may also need to take up the whole surface and put down something more permeable to allow the water to soak into the ground instead of just running across the surface. This would, in effect, make your whole yard the "dry well" and allow it to absorb significantly more of the rain water before it even makes it to the house. You'd need to talk to a local landscaper (or two or three) about your various options and quotes.
Another option, particularly if your property is the low spot in the neighborhood, is be to install "drainage tile" which is (these days) a perforated hose that water will weep into, then gives it a very easy path to follow away from the house or at least to a specified dry well/cistern that will control it and allow you to pump it from there.
I live in the Midwest in farm country. Every year in the late fall, after the harvest, I'll see huge trucks & trailers out in the fields with gigantic rolls of drain tile. A trencher digs a trench and pulls the tile in, then the dirt is pushed back over it. They work from the low spots to the drainage ditches to help eliminate the flooding that kills crops.
Of course, you won't need the massive machinery, but you'll probably end up doing something similar if you really want to solve the problem instead of just band-aid it.