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, except that the 2-cells in the squares are lax 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.)

The second definition looks like the 'lax comma category' $C // T$, where a morphism $f \to f'$ is given by a 2-cell $f \to f'\phi$. The defining universal property is the same as for comma objects, except that the 2-cells in the squares are lax 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'.

In more detail, Gray's 2-comma categories come from (strict, I think) 2-functors $A \overset{F}{\rightarrow} K \overset{G}{\leftarrow} B$. An object is a 1-cell $FA \to GB$, a morphism is a square with a 2-cell in, and a 2-cell is given by a pair of 2-cells in $K$ that fit into a commuting cylinder (it's pretty obvious if you draw a picture). In your example, (what I've called) $C // T$ has 2-cells $(\phi,\phi^\sharp) \Rightarrow (\psi,\psi^\sharp)$ given by 2-cells $\alpha \colon \phi \Rightarrow \psi$ such that $\psi^\sharp \circ f'\alpha = \phi^\sharp$. (Again, pictures make it much clearer!) So your slices are actually 2-categories, coming from $C \overset{1}{\rightarrow} C \overset{T}{\leftarrow} \bullet$.