Is this a correct definition of a line integral? This comes from the beginning chapter on line integrals in the book Mathematical Methods for Science Students:

Suppose $y=f(x)$ is a real single-valued monotonic continuous function of $x$ in some interval $x_1<x<x_2$. Then if $P(x,y)$ and $Q(x,y)$ are two real single-valued continuous functions of $x$ and $y$ for all points of C, the integrals $$\int_C P(x,y)dx,\quad\int_C Q(x,y)dy$$ and, more frequently, their sum $$\int_C \Bigg\{P(x,y)dx+Q(x,y)dy\Bigg\}$$ are called curvilinear integrals or line integrals, the path of integration C being along the curve $y = f(x)$ from A to B.

Is this a correct definition of a line integral?
It doesn't appear to resemble the wikipedia definition of a line integral over a scalar field, which makes sense to me:

For some scalar field $f : U\subseteq R^n → R$, the line integral along a piecewise smooth curve $C \subset U$ is defined as$$\int_C f ds = \int^b_af(r(t))|r'(t)|dt$$
  where r: [a, b] → C is an arbitrary bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C and a
  
  The function f is called the integrand, the curve C is the domain of integration, and the symbol ds may be intuitively interpreted as an elementary arc length. Line integrals of scalar fields over a curve C do not depend on the chosen parametrization r of C.

 A: The first is a line integral over a vector field (presented quite horribly), defined as
$$\int_a^bF(r(t))\cdot r'(t) dt$$
$F$ represents the formula for the field, and $r$ represents the path. If we write $r$ as $(x(t),y(t))$, then we get that $r'(t)=(\frac{dx}{dt},\frac{dy}{dt})$
If $F(x,y)$ represents the equation defining the vector field, we can write it in component form as $F(x,y)=(P(x,y),Q(x,y))$. so $P$ defines the $x$-component of the vector field at each point $(x,y)$, and $Q$ does the same for the $y$-component. Expanding the dot product:
$$\int_a^b P(x,y)\frac{dx}{dt} dt + Q(x,y)\frac{dy}{dt} dt$$
$$\int_a^b P(x,y)dx + Q(x,y)dy$$
Gives your version.
This is useful for determining the work done by a "field type" force (an electromagnetic field, for example) on a moving object.
The second is a line integral over a scalar field. This is just the normal integral really, just extended so it can be defined over any curve, and not just the $x$-axis in the 2D case which is rather limiting.
We call them both line integrals (since we integrate over a curve), but one is over a vector field and the other is over a scalar field, leading to different definitions.
A: The two "line integrals" you are referring to represent completely different things: Let 
$$\gamma:\quad t\mapsto {\bf z}(t)=\bigl(x(t),y(t)\bigr)\qquad(a\leq t\leq b)$$
be a curve in the $(x,y)$-plane. A graph $x\mapsto\bigl(x,f(x)\bigr)$ $\,(a\leq x\leq b)$ is a special case of this.
When a force field ${\bf F}(x,y):=\bigl(P(x,y),Q(x,y)\bigr)$ is given then the integral
$$W:=\int_\gamma {\bf F}\cdot d{\bf z}:=:=\int_a^b {\bf F}\bigl({\bf z}(t))\cdot\dot {\bf z}(t)\ dt\ ,$$
resp., componentwise
$$W:=\int_\gamma(P\,dx+Q\,dy):=\int_a^b\Bigl(P\bigl(x(t),y(t)\bigr)\,\dot x(t)+ Q\bigl(x(t),y(t)\bigr)\,\dot y(t)\Bigr)\,dt$$
denotes a work done when the force field moves a cart along $\gamma$ against friction.
On the other hand, when a scalar field $f(x,y)$ is given then the "line integral"
$$H:=\int_\gamma f\  ds:=\int_a^b f\bigl({\bf z}(t)\bigr)\ |\dot z(t)|\ dt=\int_a^b f\bigl(x(t),y(t)\bigr)\,\sqrt{\dot x^2(t)+\dot y^2(t)}\ dt$$
denotes a quantity which is related to arc length: Assume that $f(x,y)$ denotes some sort of humidity at the point $(x,y)$. Then $H$ is essentially the total amount of fluid contained in the (physical) thread $\gamma$.
The essential point of the idea of a "line integral" is that the values $W$ or $H$ only depend (a) on ${\bf F}$ resp. $f$ and (b) on the curve $\gamma$ as a "geometrical object", but not on the chosen parametrization of $\gamma$.
