Moving the primary home

[Pandu Chimata]

Hello team, I currently own a house and been living since 13 years. Its time to downsize as kids moved out of the house. 

If I buy another house/condo, smaller than I live currently , would that be considered as a primary home? How does the lender treats my next home as a primary home and what kind of documentation I need to provide for that?

Another question is - If I buy that house or condo in cash (off market for a quick close), and then can I turn around get a mortgage as a primary house, obviously keeping 20 to 30% equity in the house? 

Please help me address the above two questions. Appreciated. 

[Pandu Chimata]

Quote from @Jason Potrzeba:

Fannie Mae uses the term "Delayed Financing" to describe paying cash for a home and then applying for a mortgage to recapture some of those funds. You should have no issues if you apply and close within the first 6 months of making your new home purchase.

Here is info from Fannies sellers guide...
What is required for a delayed financing exception? (fanniemae.com)

Thank you! I assume there will be a difference of interest in taking the purchase loan vs "cash out refinance or delayed financing". Is that right?