Not knowing which university you will be employed at, I looked at several university websites regarding protocol for new employees requiring an H1B visa, including Boston University, UC Berkeley, and Missouri University, among others.
What I have learned is they have an International Student and Scholar Office or something similar, that petitions the US Citizenship and Naturalization Office for the H1B visa on your behalf. They supply information regarding you background, qualifications for the proffered position, and the position begin and end dates. They subsequently receive the approval form, Form I797, and send you copies of the petition, approval Form I797, as well as any information you might need navigating the process for obtaining your visa at the US Consulate or Embassy in your country, as well as information about traveling to the US.
Have you received these forms? If not, I would immediately get in touch with your contact person at the University and inquire when you will be receiving the forms, and if there's any other information they may be able to provide to streamline the process. They have advisors dedicated to international students and scholars and you may already have one assigned. I found departments dedicated to international students and scholars on the university websites. If you don't believe you have been assigned an advisor, you should check the school website and find that department.
From what I have read, the process to obtain the H1B Visa can take several months when everything is going smoothly; it can take much longer should an issue arise. The sooner you can get it done, the better.
When reading the information provided, it does sound like there may be an employment begin and end dates as well as contract begin and end dates provided on form I797. The employment dates may differ from the contract dates. I can only guess, maybe the contract runs by fiscal year, or maybe there's an onboarding type class you have to participate in prior to having your contract become effective, you have a different employment designation until then. It doesn't sound like a critical issue, just something needing clarification.
I hope this helps. Hopefully, if you have not yet received it, you will receive the petition, the approval, and an employee information packet that explains everything very soon.