I was wondering if options are considered a form of insurance? Is that
the right way to think about options?
Yes, an option is a form of insurance. There are differences between insurance and options, but the existence of counterparty risk is not one of them (see my comment to EnergyNumbers's answer).
Perhaps the most important difference is that options don't require the possession of (or even the intention to possess) the underlying asset, whereas in insurance the insurable risk relates to an item in possession of either the policy holder or (in the case of liability insurance) of a third-party.
When a person has a stock, buying a put option is equivalent to purchasing an insurance policy. In options, the underlying asset is a share, whereas in insurance the underlying asset is the property insured (auto, home, boat, etc). Both insurer and the writer of an option have an obligation to indemnize the buyer of the policy or option, accordingly, in the event that the underlying asset losses value.
One difference is that the nature and implementation of options obviate processes that typically take place in an insurance context. For instance, the insurer scrutinizes whether the insured incurred moral hazard (which would lead to rejecting the insured's claim), committed fraud (which would void the policy), and what the circumstances of the loss were (for purposes of subrogation and recovery from third parties). By contrast, with options it suffices to ascertain the price of the underlying asset, the strike price, and the possession of that option, these tasks being automated at the clearinghouse on the expiration date.
Strategies such as butterfly spread combine both purchase and sale (by the same party) of options. In that type of strategies, the party is acting both as insured (in reference to the options he bought) and as insurer (in reference to the options he sold).