Business indemnity insurance is a phrase rather than a product name, and it almost always means professional indemnity: cover for the financial loss a client suffers when your work or advice was wrong. The word indemnity appears in several other places in insurance, which is why the phrase sends people to the wrong page.
What people usually mean
Professional indemnity. A client contract asked for indemnity insurance, or a professional body did, and the business searched for the phrase as written. What they need is a professional indemnity policy sized to the contract, on a claims made basis, kept continuous.
The other senses of the word
Every liability policy indemnifies you, so a broker saying the policy indemnifies you against third party claims is describing public liability. A conveyancing legal indemnity is a one off policy over a title defect. An indemnity clause in a contract is a promise between the parties rather than insurance at all. All three use the same word for different things.
Working out which one you need
Read what asked for it. A client contract naming indemnity insurance alongside public liability means professional indemnity. A conveyancer raising an indemnity policy means a title defect. A professional body means professional indemnity on its own minimum terms. The requester almost always makes it clear once you look at the wording rather than the phrase.
Questions people ask about business indemnity insurance
Is business indemnity insurance a real product?
It is a phrase rather than a product name, and it almost always means professional indemnity insurance.
What is an indemnity clause?
A promise in a contract that one party will compensate the other for stated losses. It is not insurance, though insurance may or may not respond to what it promises.
Is a legal indemnity policy the same thing?
No. That is a one off conveyancing policy over a defect in a property's title or paperwork, unrelated to professional work.