As far as the IRS is concerned, either of you can claim the child, as a dependent, regardless of who's insurance he is covered by.  You can decide between you who will claim him (this assumes you live together).  Only if you can't agree does the IRS have "tie breaker" rules

The 2nd issue is based on the rules of the insurance plan.  Does the employer's plan allow a child that is is not a dependent to be covered.  Most, if not all, probably do.  Per TomD8's answer, the Affordable Care Act (ACA) apparently requires that. 

You may be able to file as Married Filing Jointly if you live in a state that recognizes common law marriage. Currently, the following jurisdictions recognize common law marriage:

 Colorado

 District of Columbia

 Iowa

 Kansas

 Montana

 New Hampshire

 Oklahoma

 Rhode Island

 South Carolina

 Texas

 Utah

Alabama (2017), Georgia(1997), Idaho(1996), Ohio(Oct. 1991), and Pennsylvania(2005) are grandfathered for the marriages before the year indicated. Living together in a common law state is usually insufficient - you need to hold yourself out as married including owning property together, having joint bank accounts, etc. To find out your state’s rules see:

http://family.findlaw.com/marriage/common-law-marriage-states.html

<<I'm asking this because I read in her insurance memo that any health benefit that I may receive through her plan as domestic partner would be subject to income tax. Is this true?>> 

If an employer pays the cost of an accident or health insurance plan (the insurance premiums) for his/her employees, including an employee’s spouse and dependents, the employer’s payments are not wages and are not subject to Social Security, Medicare, and FUTA taxes, or federal income tax withholding.  https://www.irs.gov/businesses/small-businesses-self-employed/employee-benefits

But if the employer pays for health insurance for someone who is not a spouse or dependent of the employee, the cost of that coverage might well be taxable income to the employee. 

Health insurance benefits are not taxable income; however if the benefits include sick or disability pay to the employee, those benefits may be taxable.

This web reference has a good discussion of the topic:  https://finance.zacks.com/health-insurance-benefits-considered-income-irs-7128.html

Thanks - I understand that I'm allowed to claim the child as a dependent while his mother provides health insurance. The 2nd issue however is not dependent on the insurance plan, I did a bit more research and is called inputed income. For example:

https://www.hrc.org/resources/questions-to-ask-before-enrolling-in-domestic-partner-benefits

https://ttlc.intuit.com/community/business-taxes/discussion/imputed-income-domestic-partner/00/65878...

<<Under federal tax law, the portion of an insurance premium that your employer pays for your coverage is not taxed as income. Federal law treats benefits for spouses, children and certain dependents the same way. However, a domestic partner is not considered a spouse under federal law. As a result, if you elect to have your partner covered under your plan, you will pay income tax and Social Security payroll tax on the portion of the insurance premium that your employer contributes to your partner’s policy. If your partner is an IRS-qualifying dependent on your federal tax return, these benefits would not be taxed. >>

I guess my question is still outstanding: if the mother puts the child on her insurance plan, but doesn't claim the child as a dependent, will that be considered inputed income and subject to income tax?

This is a true & correct statement  :

I read this to mean that a child need not be claimed as a dependent on the employee's tax return in order to qualify as a dependent for health coverage purposes.

BECAUSE the definition of a dependent for IRS tax rules and for an employer's insurance benefits are not the same at all ... all they share is the use of the same 9 letter word.