Allowable current density is not a random choice. It is directly related to the temperature of the conductor and is dependent on the insulation (on the individual strand, on the turn, and on the completed coil), the method and efficiency of the cooling, the heat transfer properties of the surrounding magnetic materials, and the nature of the current itself (AC or DC).
As a really rough approximation, a current density of 3000 ampere / mm^2 equates to something like a temperature rise of 80 C. A density of 4000 ampere / mm^2 equates to roughly 105 C rise. Note that this is for a machine where the primary coolant medium is air, and the basic ingredients to the insulation system are a combination of resin and mica.
Note that this is relationship is not directly between current density (measured as CURRENT PER CROSS-SECTIONAL AREA OF A SINGLE TURN) - and conductor diameter. You'd need to be able to get from cross-sectional area to diameter, and also to figure out how many conductors are necessary to create a single turn in the winding.
The equation to get from diameter (in units) to cross-sectional area (in same units^2) is:
- (diameter) = 2 * [ ( area ) / 3.14159 ]^0.5
The basics of the system design - whether that is a coil in a slot or cabling in a raceway - decide the starting point for allowable current density (or current-carrying capacity). This includes all of the following items (and probably a few I've forgotten to mention): the ambient conditions, the properties of the conductor material, the type and material of insulation, the cooling method, the primary (and sometimes secondary) coolant material, and the heat transfer characteristics associated with ALL of them.
The bottom line is that current density is not - from a manufacturer's point of view - the driver for obtaining a specific design point. It is a reflection of other choices made in the design, and serves as a check on the efficacy of the design (based on historical data).