The key in any insurance is who does it cover and what is the coverage for as that is the basis upon which an insurance will decide to underwrite the policy and it’s also a basis upon how they will price the insurance coverage. There is a second issue that also comes into play in these decisions for the contractors which is the law of Agency. Under the law of Agency, a principal may be liable for the acts of their agent, whereas an agent will never be liable for the acts of the principal. Under agency the owner or client would be considered the principal in the relationship and the supplier or contractor would be the agent.
An agent can wind up being liable or having their insurance company be liable in two situations. One of those situations is if the agent agrees to a broad general indemnification in their agreement where they agree to indemnify the principal against all claims, including those resulting from the principal’s sole or partial negligence. The second way would be if there a commitment that the principal was jointly insured.
Naming the principal an “additional insured” would allow them to claim against your insurance. Since your insurance only covers your negligence, an insurance company may agree to make the principal an additional insured as they are assuming no new or additional risks.
“Joint insured” means claims could go against either your or their policy. Insurance companies will reject the "joint insured" approach, as they do not want to be assuming claims for the negligence of the principal. They know what risks they are assuming in underwriting you, they don't know what risk the client may bring and simply do not want to be insuring them either by broad indemnifications or joint insured requirements. In fact a joint insured approach really provides the agent nothing. A party that’s injured by the negligence of the principal can’t sue the agent, they need to sue the principal. So making it “jointly insured” provides no value to the agent. What it would do is allow the principal to claim against the agent’s insurance for claims involving their negligence.
As claims against insurance affect your future premiums, you always want to avoid agreeing to language that would increase your potential exposure. I would advise you to follow the guidance of your insurer. This means that even if your insurance company would agree to provide insurance against an indemnification covering the principal’s negligence or if they would agree to “joint insured”, as a supplier or contractor you shouldn’t want to agree. If you agreed their would have pushed both the risk and cost to you and if there were claim, those would impact your future cost of buying insurance.