Whoever is listed as a tenant has the right to live in the house. Completely insane. Run, do not walk, from this certifiably crazy landlord and her loopy deal.
I agree with the above answer. In Illinois, the applicable law for landlord-tenant relationships is the written lease itself and state and local statutes and ordinances.
In addition to having a lawful right to occupy the premises, the landlord would also be shielding herself from the tenant protections provided by the Chicago Residential Landlord-Tenant Ordinance ("RLTO"). The RLTO is a very tenant-focused progressive law that provides Chicago tenants with rights that usually do not exist in other parts of the state or country. There is an express exclusion in the RLTO for owner-occupied buildings with a certain number of units. It is likely the arrangement your landlord wants would remove you from RLTO protection.
Also, written leases in Illinois are essentially contracts that are interpreted and applied with Illinois contract law. It is a very basic rule in the law that a person cannot contract with themselves. By wanting to be listed as a tenant, the landlord is essentially "contracting with herself" and very likely invalidating the lease.
You do not want to enter into a landlord-tenant relationship with no written lease protections and no RLTO coverage. I also would recommend walking away from the proposition.
NOTE: This answer is not intended to be legal advice and should not be construed in that way. This answer does not create an attorney-client relationship and no such relationship may be created absent a signed retainer agreement. The author is licensed in Illinois only, and his answer is for educational purposes alone.