The second definition looks like the 'lax comma category' $C // T$, where a morphism $f \to f'$ is given by a 2-cell $f \to f' \circ \phi$.  The defining universal property is the same as for [comma objects](http://ncatlab.org/nlab/show/comma+object), except that the 2-cells in the squares are [lax](http://ncatlab.org/nlab/show/lax+natural+transformation) natural transformations.  Your first definition should be the oplax version.

See Kelly, _On clubs and doctrines_, LNM 420, or Gray, _Adjointness For 2-Categories_, LNM 391, who calls these '2-comma categories'.

(Apologies for terseness, it's late on a Friday and this is my first ever MO answer.)