As I understand it, premiums for employer heath plans are taken out of costs so they are inherently tax free. In other words they are not consider "income" for the employee.
Correct, a properly set-up employer medical plan is a business expense to the company and employee contributions are made pre-tax.
For a self-employed person, they are allowed to deduct the entire cost of their health care insurance, so esssentially it is equivalent to the regular employer case.
Correct, however, the ACA stipulates that employers cannot deduct the cost of employee individual plans (many states already had regs like this before the ACA). Many small businesses used to simply buy, or reimburse part of the premium for, individual policies for their employees, that's now forbidden federally. So self-employed one person operations can write off medical premiums but cannot buy or subsidize individual coverage for employees.
However, a private person who buys health insurance they can only do so with after-tax income, so they cannot deduct their health insurance the way a self-employed person can. Is that right?
Correct, unless your medical expenses exceed 10% of income and you're itemizing deductions.