Difference between toolbar and interactions
Interactions are the actions available by using the mouse to zoom/pan/rotate etc the axes without activing a tool in the axis toolbar. For example, after generating a figure that supports zooming you can use the mouse scroll wheel to immediately zoom into the axes. The toolbar property is independed of the interactions property. For example, you can remove panning from the toolbar but still allow for interactive panning by mouse actions.
I agree that these differences could be clearer in the documentation.
Specify toolbar tools
To specify toolbar tools, use the axtoolbar function. This example reduces the toolbar to the pan tool.
See the toolbar link I provided above for all toolbar options.
Specify axis interactions
To disable all tools except panning, set the Interactions property of the axes. This example reduces interactions to panning.
ax.Interactions = panInteraction;
See the interactions link I provided above for all interaction options.