I'm a bit confused about the concepts of convergence rate and convergence order. Let me first give you the definitions we use: [sorry for the English, it's all self translated]
Let $x^{*}$ be our solution.
Definition 1: The sequence $x^{(k)}$ is called linearly converging towards $x^{*}$, if $$\exists L<1 \text{ so that } \|x^{(k+1)}-x^{*}\|\leq L\| x^{(k)}-x^{*}\|,\quad \forall k\geq k_0.$$
We call the constant $L$ rate of convergence.
Definition 2: The order of convergence of a numerical method is $p$, if: $$\exists C > 0 \text{ so that }\|x^{(k+1)}-x^{*}\|\leq C\| x^{(k)}-x^{*}\|^p,\quad \forall k\in \mathbb N \quad\text{with $C<1$ for $p=1$}.$$
Note: We assumed, that chose the starting value so that we get an converging sequence.
Questions:
Question 1: Can someone explain me the difference between $C$ and $L$ here?
Question 2: Can someone explain me the concept/idea behind the rate/order of convergence? [Just so I heard it from another perspective]
Question 3: Also, I often see that we use linglog and loglog plots, but I don't really get why we do that. E.g., if we have linear convergence, we can see a linear function if we lin-log plot our errors. (Why do we need the lin-log plot here)