0. The internalizer tells the brokerage that they want to buy 2000@$10.03.
1. Customer places a buy order 2000@$10.03.
2. Brokerage looks at the internalizer and at the NBBO. Brokerage realizes the NBBO is better than the internalizer and so routes a 700@$10.02 to the public markets as per their legal obligation due to Reg NMS.
3. Brokerage routes the remaining 1300 shares to the internalizer, where they are filled at $10.0290.
Customer trades are 300@$10.01, 400@$10.02, 1300@$10.029, resulting in (as you noted) a price improvement to the retail customer.
The only this doesn't benefit the customer is if (as kasey_junk notes) the brokerage is using an incorrect NBBO. If you believe a brokerage is doing this please report them to the SEC for violating Reg NMS.