What kind of insurance should an independent auto tech have to cover their liability?

First, whether or not you'd be covered under the dealer's garage owner's policy should be easy to determine: review the policy. Insurance policies are contracts; they provide exactly that coverage, and cover those people--and only that coverage, and those covered persons--as the policies state they do. Ask the dealer if you can look at the policy; if you have any questions, ask if he'll authorize his insurer to speak with you.

However, even if you are covered under his policy, treat that as additional coverage--you don't want to rely on coverage not under your control. Therefore, you definitely want your own policy. You'll want a commercial or business policy with  good umbrella coverage over it, to cover your tools, inventory, any professional negligence (e.g. bad repairs), and for any other liabilty, such as if someone is injured. You can also get business interruption coverage, to provide income to you if fire, theft, etc. forces you to shut your business down temporarily.

Speak with a reputable insurance agent--he or she can recommend the coverage you need. It does not have to be expensive: in the past, I owned a small publishing company which had three employees, $150k of inventory, and did $500k of sales, and my policy cost less than $2k per year. (Which was also deductible as a business expense.)

You shojuld also set up an LLC or limited liabilty company for your business. An LLC will  insulate your personal assets from most business-related debts or liability, such as for personal injury (unless you personally hurt the other person; then you could be sued as the at-fault party), breach of contract, loans (unless you personally guaranteed them), bad repairs or auto damage, etc. An attorney can set one up for you for a few hundred dollars. As an added bonus, an LLC, by separating personal and business expenses, makes it easier to take business expenses as tax deductions.