Each policy type is different. When the nature of the thing to be insured changes, you need to talk to an agent or insurance broker.
If some damage occurs, will the normal home insurance cover absolutely
nothing for me because I should've had landlord's insurance?
Lets say you have standard homeowners insurance, and something vital to habitability breaks. Your policy might or might not cover the replacement. But as a landlord you have an additional issue: state rental law may require you to refund part of their rent while the item is being fixed; you might have to to put them in a hotel while the repair is made. The standard homeowners policy might not cover the added expenses for this unspecified person, and the loss of income for your business.
This is from the website of a US insurance company:
Heating or Air Conditioning Loss Reimbursement provides reimbursement
for payments made to a tenant as a result of the complete loss of
heating or air conditioning to a tenant's leased unit due to covered
mechanical breakdown or electrical failure of the heating or air
This isn't covered in a standard homeowners policy.