Start with this. If you have occurrence coverage and always have had occurrence coverage, you're fine. In other words, if you get hit with a claim, you're covered by whoever you were paying for the occurrence policy when the incident occurred, hence the name "occurrence" coverage. It doesn't matter who you work for for when or where, if you were paying for occurrence overage, when it occurred, your covered indefinitely, no matter where you go, who you end up working for and in what capacity.
Claims made is covered by whoever you're paying for insurance when the claim is made, not when the incident occurred. So, if you're at the same job, and same malpractice company when you get hit with a suit, you're covered. If you leave jobs and change insurance then under claims made, you therefore detach yourself from that insurance company and they no longer cover you. Therefore, you've then got to pay someone to cover you until the statute of limitations is up, ie, cover your "tail."
Why? The new claims made company isn't going to cover you for three years of patients and associated liability after paying one day, one month or one quarter's worth of premium. That would be a losing proposition. That's like expecting to by hurricane insurance and getting a new homeowners insurance company to cover you for any hurricane that hit you in the three previous years, and future years.
Also, don't tie the malpractice insurance policy to your job. The coverage has nothing to do with your job. The coverage is for you, not your job. They may write the check to pay for your policy (out of the monies they would otherwise have been available to pay you, ie, out of the physician salary pool) but the insurance company doesn't care where you work. They just care that the policy is being paid for by you, or by someone on your behalf, and that you're practicing within your scope of practice, ie, emergency medicine. They don't care where.
The difference in the types of coverage all comes down to the statue of limitations, 3 years in many states. Who will cover you during that 3 year trailing delay where claims could still hit? Occurrence will. Claims made won't (allows them to advertise and charge a lower rate). But they will, if you buy extra coverage to cover your "tail" for those 3 statute-of-limitation years.
Think of it like auto insurance. You only need one policy to cover you for driving. You don't need to buy separate car insurance each time you drive your friends car, a rental car, etc. Nor do you have to get a new policy every time you drive on a new road or in a new state.
All that in a nutshell = Get occurrence and no worries. Get claims-made and you're paying for tail coverage any time you switch to a new policy.