The type of home insurance policy that one buys does change depending on the type of ownership. However, the occupation of the property is also a key factor.
If the owner of a freehold property lives there themselves, they'd buy a homeowner's policy. If they rent it out to others, they'd buy a landlord insurance policy and their tenant would buy a tenant insurance policy.
When it comes to leasehold property, most leased residential properties are ready-to-live-in homes, like apartments or townhouses. There aren't many people who lease empty land to live on and build their own house.
Accordingly, most people who lease their homes need tenant insurance policies as well. They don't own the building they live in, so they don't need to insure it—their landlord (the freehold owner) does.
Finally, condo owners need condo insurance policies. Condo insurance policies have features unique to condo ownership, mainly that they're meant to work with the condo corporation's master insurance policy.
If you're unsure about what type of insurance you need for your home, get in touch with your insurance provider. They'll be happy to help you figure out the best policy for your ownership and occupation status.