I've never heard of Fawn being reliant on silver being present. Do you have a source for that info? Everything I'm finding is indicating that Fawn is just the common name for dun. 
 
Beautiful birds! Just as a means of helping you understand the differences between them, Fawn, and Platinum, do you know the genetics for the Mauve coloration? That they are solid Black (two copies extended black, E/E), plus Blue (one copy of the blue gene, Bl/bl+) and Chocolate (two copies of the chocolate gene in males, choc/choc, but one copy in females, choc/-, because it is a sexlinked recessive gene)? Platinum is a similar variety, but based on dun instead of Chocolate, so you would have a Black bird as the base with Blue and dun (I^d/i+). For 'just' Fawn, take the Blue out of the equation of the Platinum bird, leaving a Black base plus dun (and... maybe silver as well, I guess?).
It's kind of hard to be any clearer than that. I can try to explain a bit, though, to maybe help you understand a little bit about what all those symbols mean. When you see an 'equation' like what the other user posted, or letters like what I have in the above paragraph, what you're looking at is basically a code to explain the genes. These genes come in pairs, one from the bird's father and one from their mother. Capital letters mean that gene is dominant (for example, E/E in my above paragraph), while lower case is recessive (as in choc/choc). Any time you see a plus sign, that means that the gene it's following is wildtype, or that that's the gene that occurs normally in the ancestral wild population (like Bl/bl+, where Bl is the Blue gene and bl+ is the wildtype of not being Blue). If there are no differences from the wildtype in a pairing of genes, it's generally not mentioned because it's understood to be the same as wildtype. For example, I wouldn't mention that your birds are not Lavender or split (as far as I'm aware), and therefore are Lav+/Lav+, because that is understood to be the case. Lastly, if a gene is sexlinked, then the males may have two copies of it (again, as in choc/choc), but females will only ever be able to carry a single copy (as indicated by the dash in choc/-).
It gets a bit more complicated when you have many different alleles of the same gene. That's where you may see a superscripted letter, or one with a ^ and a letter, as in the dun gene, I^d. All that means is that there are several different things that can fit in that 'slot,' dun being one of them, but you CAN'T have both I^d/i+ AND I^s/I^s in the same bird, as an example, because there's only enough room for two alleles in that slot (so either I^d/i+ or I^s/I^s, or some other combination of two).
Sorry, I hope that wasn't even more confusing! I'm never sure if I'm being too wordy or as clear as mud when I explain things like this. 

 As for the confusion with the calculator, I've played around with it, but not as much as others, so I don't think I can explain the discrepancies between what it says and what should happen.