All of what you mention is absolutely correct, except to be clear...your bodily injury coverage pays for other person's bodily injury that you are liable for. Whereas, Uninsured Motorist (BI) pays for your bodily injury if the liable party's liability coverage is not enough.
Uninsured Motorist or Underinsured Motorist coverage will reimburse you, a member of your family, or a designated driver if one of you is hit by an uninsured or hit-and-run driver.

Underinsured motorist coverage comes into play when an at-fault driver has insufficient insurance to pay for your total loss. This coverage will also protect you if you are hit as a pedestrian (in some states).

You may also consider purchasing an umbrella or excess liability policy. These policies pay when your underlying coverages are exhausted. Typically, these policies cost between $200 and $300 per year for a million dollars in coverage. If you have your homeowners and auto insurance with the same company, check out the cost of coverage with this company first. If you have coverage with different companies, it may be easier to buy it from your auto insurance company. You can comparison shop your policy here by reviewing assurance voiture quotes.

We have partnered with industry leaders to help you find the best rate for Umbrella insurance. Safeco's Personal Umbrella policies are competitively priced and offer exceptional coverage. You can receive coverage up to $2 million.

Umbrella policies are available to Safeco customers who insure their personal autos with Safeco. You may have to have certain liability limits in order to qualify for Umbrella insurance (availability is subject to state laws, compliance and underwriting).

Safeco also offers homeowners policy which you can read more about at this link or by calling 1-800-936-0356.