You can have both, they are not exclusive of each other. Addisons or the better name Adrenal Insufficiency is when not enough cortisol is made by the adrenals to sustain life. And as you mention, it can get low enough to cause symptoms, without being to the stage of life or death.
True Addisons is usually either autoimmune (there is a test), from having had TB, having something physically wrong with the adrenal gland (cancer/removal). There do seem to be some that have Addisons/Primary or type 1 Adrenal Insufficiency that there adrenal glands just stop working. This seems to be with complex patients with comorbidities going on. The pituitary is sending out the “signal”, ACTH, to the adrenals, but the adrenals do not respond. Usually ones ACTH levels will be elevated because the pit will continue to send out more and more ACTH trying to get the adrenals to respond and send out cortisol. Signs of high levels of ACTH are hyper pigmentation, dark blotches on the skin, looking tan/golden when you haven’t been in the sun. Common to type 1’s are fluid imbalances, not being able to hold onto fluid you take in (just pee it right out)needing fludricortisone to be taken along with the cortisol replacement, Hydrocortisone (pills or injectable - not the skin cream).
Secondary Addisons is when the pituitary is the problem. The pituitary is not sending out the signal, ACTH, to the adrenal glands, so the adrenal glands have no idea they are supposed to be making cortisol. ACTH testing will show low to no ACTH. Pituitary’s can be damaged in many ways, also autoimmune issues, tumors (very tiny), huge blood loss (such as in Sheehan’s Synyndrome). Usually if the pituitary is the problem there will be other hormone deficiencies (other ones the pituitary makes, which are numerous!).
Typical low cortisol symptoms can include: no appetite, nausea, vomiting, weight loss, no energy, sleeping more than normal (can be a lot more than normal), hair loss, low blood glucose levels, low BP (can get dangerously low), low or high potassium, low sodium (all electrolytes can get very messed up), etc.
Hope this helps!