Question on Ebers-Moll model
Notice the direction arrows on the diagram:



All terminal currents are taken as positive when going in to the terminal. This is a common convention in circuit theory, even if practically the emitter current will normally be flowing out of the terminal.

With this convention in mind, there's no discrepency between the two formulas and both are written in terms of conventional current. \$I_E\$ will have a negative value when the device is in the forward active or saturation modes of operation.