No. What DSP meant is that when you enter they consider you a tourist but what they technically issue is a "transitory residency" valid for 90 days. Previously they would put a stamp in your passport that said "90 days", now you just get an email. Those 90 days are your "transitory residence". You don't apply for it as such (unlike the ""temporary residency categories)--your application for the "transitory residence" is simply the act of physically presenting yourself at the immigration country with your passport to enter the country after getting off the plane.
During your 90 days of "transitory residency", you can apply for a 12-month "temporary residency", if you fit into any of the categories of "temporary residence" (e.g., the rentista category, but there are various others). To obtain a temporary residencty, you have to formally apply according to a set process and provide the documentation your category (e.g., rentista) requires. If you are successful, they issue you a 12-month "temporary residency". After 12-months you can apply to renew it. After another 12-months you can apply to renew it again. At that point you will also have been here long enough (2 years, 24 months) to apply for "citizenship" (an entirely separate category and process to "residency"). You can also, at that point, apply for third year of your "temporary residency" and at the conclusion of that third year of "temporary residency" will now be qualified to (separately to any citizenship application process you may have commenced at the 2-year mark), apply for "permanent residency."
