Effective or minimal area of core to calculate losses?
The minimum area of the core's cross section can point to a situation where flux density saturation problems can appear if you have too many ampere-turns. Regard this as a potential bottle-neck point for the magnetic lines of flux.

The effective area of the core's cross section should be used to calculate the inductance for a given number of turns (or use the \$A_L\$ figure instead). This is the proper figure of merit for most inductor calculations: -



Notice the top two formulas contain the term \$\Sigma\dfrac{l}{A}\$

That term above equals the effective length of the core divided by the effective area of the core and is portrayed this way because it implies the core is made up from a series smaller lengths (\$l\$) in ratio to the core area for that particular length. It's a way of describing the core shape (in terms of magnetics) as one figure.