Let $(M,g)$ be a Riemannian manifold. $M$ is called complete, if the maximal geodesics on $M$ are defined on all of $\mathbb{R}$.
The injectivity radius of a point in $M$ is the largest radius, s.t. the exponential map is a diffeomorphism.
The injectivity radius of $M$ is the infimum of the injectivity radii of all points.
Now in some sources, I have read that completeness and non-zero injectivity radius are equivalent conditions, while in other sources, it says that completeness implies non-zero injectivity radius (but not vice versa).
I however don't really see an obvious connection between these two definitions..
Example. The set $M \subseteq \mathbb{R}^3$ described by the equation $x^2+y^2=e^{-z}$ is a $2$-dimensional submanifold. It's complete, but the injectivity radius is $0$, as can be seen by looking at the circles $z = \mathrm{const}$ (and large), which get arbitrarily small.
Using the definition you provided, you should be able to show that positive injectivity radius implies completeness. If this radius is $r>0$, one can extend any finite geodesic at least by $r/2$.