Project support can be messy
Project plans generally fall under the broader category of project support materials. Their nature and structure depends on the project, you, possible collaborators and other factors. The tools you use can have a big effect: someone using omnifocus has options that someone using paper does not. Roughly speaking, I think of projects as having different levels of complexity:
Really simple: there's going to be an obvious next action at every point, and I don't need any plan. An example is freezer repair, where the first step is to find possible repair companies, and go from there.
Simple: some sort of simple plan is needed. I generally will use a note attached to the project entry to store any thoughts about issues, major components, et cetera. About 90% of my projects are simple or really simple.
Complex: this takes lots of forms. A project can have a simple linear structure, but milestones, multiple people involved, et cetera requires more tracking. A project may have many sub-projects that can be worked on in parallel. The nature of the project may have multiple phases, and the first phase may be R&D to determine the nature of the project. A number of tools can be used: text documents, mind maps, outlines, project planning software, et cetera. The natural planning model is valuable for any project when more clarity or more action is needed, but it is especially helpful for complex projects. The example project you gave seems suited for its own document in Google Docs, which you can refer to as necessary or desirable.
You also mention all the other projects and commitments you have. GTD objectifies projects, next actions and calendar items, and this can seem overwhelming. You have the commitments with or without GTD, but the outcomes will be better with GTD. I cannot stress enough the value of the weekly review in this. Good luck!