You are asking great questions and have done a lot of research!
Your figures look right with 2 exceptions. First the "$18,000" in allowable costs should be $16,000. The math in your second scenario (18,000 x 0.5 = $8,000) gets to the correct credit amount. Your math in the first scenario is off though. $16,000 - 10,500 = 5,500 x 0.5 = $2,750.
The more important issue you did not consider is the effect of the pre-tax contributions to your FSA. If you contribute the full $10,500 and your income is in the 22% tax bracket, your tax liability will be approximately $2310 lower. You also have another $800 is savings not directly related to your tax return since your FSA contributions are not subject to social security or medicare taxes. That being said, you still come out with a lower overall tax bill using option 2.
Please note all of this is based on the information you provided and assuming that you are filing a joint return with AGI under $125,000.
