Thanks El Manolo.You can only apply via DNM website if you are physically present in Argentina. Maybe you didn't use the correct term in your conversation with the Embassy. You would apply for a visa, a family reunification visa if you are in Spain.
So, I should apply for Argentinian permanent residency when I am physically on the ground there.
Conversely, I apply for my wife's permanent residency from here in Spain. I have read the small print - I first need to physically present the required documents here. Thereafter, for step 2, she needs to visit the local Spanish Consulate in Mendoza.