How do you do a surface cut without cutting the entire object
Two options include:
1. Since you already have the sketch lines made, use them to split (Insert - Curve - Split Line - Projection) the model faces. Then select the unwanted faces, right click, choose delete, and use the fill option.
2. If you had not already laid out the sketch lines, I'd suggest using a vertex chamfer.
You'll do this step 3 times, once for each sketch. Insert - Curve - Split Line - Projection
Once all three faces are split, select the unwanted faces, right click, choose delete face and turn on the option to fill.
A chamfer on a vertex is a good solution as well.
If you really want to use a cutting option with your reference plane, the Split tool will do this task. Insert - Feature - Split. Just be careful of which bodies you choose to keep. Often with this method I'll just keep all of the bodies, and then delete or merge the results in a separate step.