Defintion of Solubility in Different cases
There are two ways of dealing with this. 

Solubility product

The more general way is to state the solubility product. Taking your first example, this would have the same value no matter how much acetate is present in solution before adding the silver ions. The disadvantage is that for highly concentrated solution (in fact, in the presence of ions, even for moderate concentrations) this gets inaccurate unless you switch from concentrations to activities.

Solubility (of a solid)

The other way is to specify the solubility of $\ce{AgCl}$ in pure water, in 0.1 M sodium acetate, in 0.5 M sodium acetate and so on. In this case, because silver chloride is much less soluble than silver acetate, no silver acetate would precipitate (the silver ion concentration will be sufficiently low in solution so that silver acetate will not be exceeding the solubility product).

Similarly for AgCN, you could specify its solubility in pure water, in aqueous solution buffered at pH 7, or buffered at any other pH. For the latter example, you should probably also specify ionic strength or the entire makeup of the solution prior to adding AgCN. In this case, there will be a strong dependence of the solubility on the pH because HCN is soluble and its formation will decrease the concentration of cyanide ions while the concentration of silver ions will remain the same up to the point where it precipitates.

In both of these cases, there is no ambiguity in terms of concentration because there are no sources of the ions forming the precipitate other than what was added as a (stoichiometric) solid.

On the other hand, if you add NaCl to a solution already containing LiCl, the apparent solubility of NaCl will be lower than in pure water because of the common ion effect. You could still talk about the solubility of NaCl in 2 M LiCl, and the quantity would refer to the concentration of sodium and chloride ions due to the addition of NaCl, not due to ions already present before addition.

Concentration of ions vs. concentration of added solid

With any salt that does not have 1:1 stoichiometry, we give the concentration of the salt, not that of the ions. For example, we talk of a 1 M $\ce{MgCl2}$ solution even though it is a strong electrolyte containing 1 M magnesium ions and 2 M chloride ions. The concentration given is based on the formula of the solid added, not on the concentration of ions after dissolution. In a similar way, we would say we added acetic acid to a concentration of 1 M even though we know full well that some of it will be ionized.