"BTW, either you are an employee or you are an IC. You can't be both."
You are either/or - you can't be both at the same time. (Well, it is possible for an employee to undertake a separate project as an independent contractor, but that's a rare situation - and apparently not what you are talking about.) Read the IRS link that she gave you, remembering that it is addressed to your boss, the employer, not to you, the employee. Most of the risk of penalty for mis-classification is on the employer, not the employee. But it wouldn't hurt to discuss this subject with the company's attorney or CPA.
What version of Quicken are you using? Are you using H&B? Or QuickBooks? Are you running a business, in addition to this ambiguous employee/contractor engagement? Do you also work for other "employers"? It seems to me that your "invoices" are the equivalent of time sheets submitted by many employees, showing hours worked in a week or month. Or the daily time cards stamped by time clocks in many factories and other workplaces. Rather than Quicken, you could use Word or some other program to print your invoices.
Assuming that Laura and I have correctly understood your situation, then the "invoices" that you issue don't need to be recorded on your books at all, except perhaps as memo entries if you choose. You should record the payments you receive as paychecks, showing the full invoice amount as "gross pay" and recording all taxes and other deductions, and showing the "net pay" as the deposit to your bank account. At year-end, you should receive a W-2 form and the numbers on it should agree with what you have recorded for gross pay and for income tax and FICA withheld and other amounts deducted. As I'm sure you know, "gross pay" is income and that's what you'll pay tax on; "net pay" is what you can spend, but there's no place for it on your tax return.
RC