My previous reply is when the logged-in user is the project owner. Since, you are looking for team member role, follow the below approach.
Remove My Tasks category from the Team member group and add My Organization category to the Team member group. Then, set the following Permissions for My organization category to 1. Create Deliverable and Legacy Item Links 2. Create New Task or Assignment 3. View Project Schedule in Project Web App 4. View Project Site 5. View Project Summary in Project Center 6. View Resource Assignments in Assignment Views.
With the above change, the team member will be able to create deliverable and tasks updates on projects they are part of (Build team) but will be able to view all details of all projects.