Does the scrum master/lead assign tasks to the team on the task board or should the team create it for themselves in an agile approach? 

In an ideal theoretical scenario, the Product Owner prepares and prioritizes tasks in the Product Backlog. During Sprint Planning, the Development Team selects tasks from the Product Backlog to include in the Sprint Backlog. Once the sprint begins, team members assign themselves to tasks from the Sprint Backlog. In Agile, on the other hand, flexibility is allowed, so in exceptional cases, developers can adjust the Sprint Backlog during a sprint - adding or removing tasks - provided the Sprint Goal remains intact and changes are agreed upon by the team. 

Of course, theory is one thing, and practice is another. In reality, these “exceptional cases” aren’t always so exceptional. In our team, it’s not uncommon for developers to add or remove tasks during the sprint if needed (e.g., unexpected issues arise during coding, additional work is required, or some tasks won’t be tackled due to unforeseen circumstances, like the business dropping an idea). That said, we strive to maintain the team’s velocity and the sprint goal - although, let’s be honest, sometimes we don’t quite succeed.

On our teams, the team members are required to create their own tasks but they can also get created an assigned in a standup if something comes up.  What I have noticed with most big vendors we work with (Accenture, PWC, etc.) is they have a lead that creates the items and then assigns them in a sprint planning meeting before the sprint starts.