The amount of detail used in a test plan will depend upon your particular context. When I'm asked how much detail to include by clients, I've started drawing a simple 2 X 2 matrix that looks like this:
There is simply too much variation between different teams of testers and business contexts to provide a one-size-fits-all answer to this question. There are good and valid reasons that different teams around the world use very different test plan documentation approaches when it comes to test case writing styles.
You and your colleagues should familiarize yourself with several different approaches that other teams use. Think about the pro's and con's of using formalized and detailed script structures as opposed to the pro's and con's of other "documentation-light" approaches. Once you're aware of the options available, you should consciously adopt approach that works best for your context.
These sources are directly on point and provide more examples for your consideration than I can get into here:
Dr. Cem Kaner's excellent piece "What's a Good Test Case?"
A presentation I gave at an STP conference called "Documenting Software Testing Instructions - A Survey of Successful Approaches"