An eligible residence can be sold for a profit of up to $250K ($500K for married) without paying taxes (on the profit portion or the base investment). Whereas all accrual to a secondary or vacation home will be taxed upon a sale at the capital gains rate. Your taxes and interest writeoff from your first 2 homes work the same, they can be written off.
Mortgage companies may give different rates for a secondary residence, this is situational, based on debt ratio and credit rating.
The homestead exemption is a write down of the property value by a set amount that's approved by the county or state, but only on your principal residence.
So primary residence has a different meaning to the mortgage company, federal government and local government and is designated by guidelines more than strict rules. The purpose of the guidelines is to ensure you don't get tax benefits for an income property. So that's step one, if you're making money off of it, you can't claim primary for any of the 3 categories.
For taking the loan, the mortgage company can consider it your primary residence, as you're buying it for family and plan to spend a significant amount of time there and it will be your first home. Your family should suffice as fulfilling the owner occupy clause, which are in some loan agreements that the primary tenant be the owner for the first year, this is sometimes included for certain loan types as a risk reduction. There's not a separate legal distinction for the mortage company to issue a loan for a primary residence, it's just a risk calculation. Your only legal due diligence here is to try not to violate the owner occupy clause.
For the federal government, it's a little different. They don't really need to distinguish much between your first 2 houses for writeoff purposes, as long as it is not primarily a rental property. To be an eligible residence for the purposes of a sale and capital gains protection, you must have owned the home for 5 years and lived there for 730 days out of the 5 years before the sale. Notice, the IRS doesn't require or use the words primary or principal, just eligible.
For the state or sometimes by county, the implication is a little different. You must have owned the home on Jan 1 and have it as a "principal" residence according to many different states (your state may be different though, please check). For example: Texas only requires you to declare your principal residence by having it on your licence and filing your government paperwork there. And to not have another primary residence.