I don't think there's a formal definition of "extended" BF. The World Health Organisation recommends that breastfeeding continues until the child is at least two years old, so logically "extended" breastfeeding would be past that point, but as your HV has shown many people define "extended" as "the point where I personally get weirded out by it". I'm not sure the term "extended breastfeeding" is particularly helpful; I prefer "natural term breastfeeding", i.e. breastfeeding until the child chooses to stop. In the case of mine DS nursed until he was 3.2 but DD only until about 20/21 months. Both of them self-weaned very gradually.
I have never understood the "just about the mother" thing. I would love to find a HV spouting that stuff at me. Are they saying that your child shouldn't have milk at all at this age? No. Are they saying that cow's milk is more beneficial and better suited to the needs of a small human than human milk? No, not normally (and if they were I would like to see one tiny shred of evidence for that view). So in what possible way is giving a child the milk he needs in the form best suited to him "just about the mother".
Anyway, for the specific health benefits:
In the second year (12-23 months), 448 mL of breastmilk provides:
29% of energy requirements
43% of protein requirements
36% of calcium requirements
75% of vitamin A requirements
76% of folate requirements
94% of vitamin B12 requirements
60% of vitamin C requirements
(Source: Dewey KG. Nutrition, Growth, and Complementary Feeding of the Breastfed Infant. Pediatric Clinics of North America . February 2001;48(1))
Nursing toddlers between the ages of 16 and 30 months have been found to have fewer illnesses and illnesses of shorter duration than their non-nursing peers. (Source: Gulick EE. The effects of breastfeeding on toddler health. Pediatr Nurs . 1986 Jan-Feb;12(1):51-4.)
Antibodies are abundant in human milk throughout lactation. In fact, some of the immune factors in breastmilk increase in concentration during the second year. (Source: Goldman AS, Goldblum RM, Garza C. Immunologic components in human milk during the second year of lactation. Acta Paediatr Scand. 1983 May;72(3):461-2.)
One study [I only have it noted as "Ferguson et al, 1987"] showed a significant link between the length of time children had been breastfed as toddlers and their teachers' ratings of how well socially adjusted they were aged six to eight.
Increased time spent bf also offers some protection for the mother against breast cancer, ovarian cancer, uterine cancer, endometrial cancer, osteoporosis and rheumatoid arthritis.
It's also great for helping them calm down and great when they are ill and don't fancy eating.