Does a custody agreement outside of the court legal?

My ex-wife and I have joint legal/decision making and physical custody. Our current court ordered time sharing agreement is 50/50 alternating weeks. Since last July, we came to an agreement outside of court and we changed the time sharing. Nothing was submitted to court though. Since July, per our agreement, our son has been living with me, my fiancé and our 3 kids from Sunday night to Thursday night because we enrolled him in preschool in our school district and he is with his mom every weekend Thursday night to Sunday night. Since his mother couldn't afford the preschool tuition, we paid for it for his benefit. As of January, he qualified for the special education program at his current preschool and tuition is now free. Now his mother wants to enroll him in preschool in her school district starting in August. I do not agree with this for multiple reasons: 1) he is in a grade A school district for elementary and grade B for high school which is 5th on the list of top schools according to the Department of Education website itself. 2) His mother's school district is grade B for elementary and grade C for high school 3) she moved an hour away without checking out the community, school districts, etc. prior. She moved in with her new husband and his kids and states she had to move there because she couldn't afford to live on her own and her husband can't move because his kids have to be in their school district that’s the same as his ex 4) my fiancé and I work opposite shifts for the most part therefore he is not in daycare at all during the week with us and we take him to school, do homework, take him to practices, doctor appointments, volunteer in his school, etc. His mom works Monday to Friday, 8am-5pm so if he were to be with her all week for school, her quality time would be limited to only a few hours each night leaving her hardly any time to work on homework and she can't miss work to take him to doctor appointments which are during her work week.