I'll assume for the moment that you have some sort of landlord's policy that combines property and liability insurance. In that context, "Broad Form" should be a mid-range of coverage for your property. A "Standard Form" would have a limited list of covered perils (causes of damage or loss), a "Broad Form" has a longer list of perils, a "Special Form" generally does not list perils specifically but covers every type of damage or loss to your property unless specifically excluded.
That would cover damage to the property of other people if your negligence caused the damage. In your policy, your insurance agrees to pay if you are responsible due to negligence and agrees to defend you if you were not negligent.
In this case, your insurance company has denied the claim for damage to the downstairs unit because it believes (as I do) that you were not negligent. Should you be sued for the damage to the other unit your insurance company will hire a lawyer to act on your behalf.
That's the coverage for damage to the structural components of your unit. Read your CC&Rs for the definition and you'll understand that $1k is not nearly enough to cover potential structural damage to your unit elements.
That is part of the coverage for your property: your unit and any personal property that belongs to you. It does not cover the property of anybody else.
The HOA people typically tell an owner that he or she is responsible for any damage to anybody, in the hopes that the owner doesn't do his or her homework.
That's why I suggested you read your CC&Rs carefully to see if you have any contractual obligation for damage to anybody simply by virtue of you owning the unit.
Without reading your Landlord's Liability coverage part I can't give you an opinion on whether a contractual obligation might be covered or not. If you can find the exclusion or limitations on contractual liability and quote it word for word, I'll be able to comment on it.

