First of all, while there may have been a coincidence of events when you observed the problem, there is no interconnection between your well pump / supply piping and the function of the drains.
Although you did not mention it in your post, the laws of physics dictate that water will not run uphill, and given that the laundry standpipe is much higher than the top of a toilet bowl, the toilet in question must be on a floor above the laundry standpipe, such as the laundry in the basement and the toilet on the first floor, or laundry on the ground floor and toilet on 2nd floor.
There must also be a partial clog in the drain line slightly below the point where the laundry standpipe ties into the main stack. As the water from the toilet enters the line it is beging delayed by the partial clog sufficiently to cause it to rise up in the pipe above the level of the laundry standpipe opening.
The solution would be to pull the toilet and run a powered auger drain cleaner down the toilet line until you clear the clog.
You would need a machine capable of running a 3" cutter with at least a 50' cable.
While this job is often relagated to commercial drain cleaning services, it is not difficult to do and surely not beyond the ability of anyone with moderate DIY skills.
You can rent the machines from most of the large home supply stores such as Lowe's or Home Depot or your local tool rental company.
If you need detailed instructions on how to clean the drain post again and I will post it for you.
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