Let $f:X\to Y$ be a map between connected CW complexes and $k\geq 0$ an integer. I am confused by the definition of $k$-connectivity or more fundamentally by what induces a long exact sequence of homotopy groups.
My favourite definition for $k$-connectivity is this: $f$ is called $k$-connected if the homotopy fiber $F$ of $f$ is $k-1$-connected, meaning that $\pi_i(F)=0$ for all $i$ with $0\leq i\leq k-1$. Of course, this definition is only reasonable for connected spaces $X$ and $Y$.
I know that for $F\to X\to Y$, there is a long exact sequence \begin{equation} \ldots\to\pi_i(F)\to\pi_i(X)\to\pi_i(Y)\to\ldots\to \pi_0(X)\to\pi_0(Y) \end{equation} by arguments about the homotopy fiber $F$. I like to define the relative homotopy groups $\pi_i(Y,A)$ as $\pi_{i-1}(F)$ and one gets from the above long exact sequence a long exact sequence for the relative homotopy groups.
Now Wikipedia defines for an inclusion $f:X\hookrightarrow Y$ to be $k$-connected, if its homotopy cofiber $C$ (= mapping cone) is $n$-connected, meaning that $\pi_i(C)=0$ for all $i$ with $0\leq i\leq k$. Even worse for me, the same Wikipedia article asserts a long exact sequence \begin{equation}(*)\hspace{10ex} \pi_i(X)\to\pi_i(Y)\to \pi_i(C) \end{equation} (however this is prolonged to the left and to the right).
My main question is: How do the two definitions of $k$-connectivity relate?
Maybe however, my problem of understanding begins even earlier: How do $\pi_i(C)$ and $\pi_i(Y,X)$ (from de definition above) relate?
I was able to show that for the connectivity \begin{equation} conn(F)+1=conn(C) \end{equation} holds for simply connected $X$ and $Y$. This means, that for simply connected $X$ and $Y$, the two definitions of $k$-connectivity of $f$ coincide if there is really an exact sequence (*). But what happens when $X$ and $Y$ are not simply connected but only connected?