A "land contract" is not a way of renting property, it is a way of purchasing property on an installment basis without bank financing. It is Ohio's version of what in some other places is known as "contract for deed".
See "What is a Land Contract in Ohio" and "How Land Contracts Work" The actual law is Section 5313.
In a land contract, the buyer has equitable but not legal title. The buyer normally pays all taxes and fees, and is responsible for maintaining the property, just as if s/he has bought the property. But if the buyer defaults, all payments and equity would be forfeit to the seller. Until the buyer has paid 20% of the purchase price, or made 5 years of payments (whichever comes first) a single missed payment constitutes default and can lead to the buyer being evicted with all payments to date going to the seller, the buyer coming out of the deal with nothing.
Also, if the seller still has a mortgage and defaults, the buyer may lose everything paid to date. The buyer does not have the protections that a lease gives a tenant, nor the protections that legal title gives a purchaser via a traditional mortgage.
Land contracts are often used when the buyer cannot qualify for a mortgage.
The buyer pays interest, and it is often at a higher rate than the current rate on a mortgage.
Land contracts are often a form of predatory lending, but for some buyers they make sense. A buyer needs to carefully review the contract with a lawyer knowledgeable about land contracts, and consider the risks and benefits of this form of financing.
As I understand it, there cannot be a valid land contract for one apartment in an apartment building. A land contract must be for title to the land and all fixtures, including all buildings, on it. (There was at one point some unclarity if the question referred to an apartment. It is now clear that it refers to a house, so this statement is not relevant to the OP, but may be to others.)
It is not clear just what the OP's landlord (LL) has in mind. It may be that LL plans to offer a "land contract" in which the purchase would be completed only after a very long time, with the idea that the OP would simply default when s/he wanted to move. Such a default could harm the OP's credit. There seems no benefit to the OP in such a scheme compared to a lease, unless LL will lower the price significantly, taking into account maintenance costs and taxes, which OP may well be expected to pay under a land contract.
Note that a landlord can't legally force a tenant to sign a document cancelling a lease, or to sign whatever s/he will call a "land contract". Nor can s/he cancel the lease without the tenant's consent except for good cause as specified in the law (such as not paying rent). S/He could become uncooperative on other matters if a tenant doesn't do as s/he wants.
If a tenant does cancel his or her lease, s/he will lose some rights. Others are guaranteed by law as long as the tenant is paying rent. If one signs a "land contract", what happens depends on its provisions.
OP needs to very carefully consider just what is being offered, and its risks and any possible benefits. Details of the contract will matter.
No matter exactly what LL has in mind, this is not at all a usual procedure for a landlord. OP or anyone in a similar circumstance should be very careful.