Let's take a sample reaction $\ce{\it aA + bB-> pP + qQ}$ with stoichiometric coefficients $a$, $b$, $p$, and $q$ for reactants $A$ and $B$ and products $P$ and $Q$.
The rate of reaction is the rate of change of the extent of reaction. Extent of reaction is usually given the symbol $\xi$, so reaction rate is $\frac{d\xi}{dt} = \dot{\xi}=r$.
If the reaction rate is $\dot{\xi}$, then the rate of change of concentrations is:
$$\frac{d[A]}{dt}=-a \dot{\xi}$$
$$\frac{d[B]}{dt}=-b \dot{\xi}$$
$$\frac{d[P]}{dt}= p \dot{\xi}$$
$$\frac{d[Q]}{dt}= q \dot{\xi}$$
That is the stoichiometric coefficients link the rate of formation/consumption of the products/reactants, sort of by the definition of stoiciometry.
Now, when talking about the order of a reaction, what we are doing in effect is giving formulas (or partial formulas) for $\dot{\xi}$. If the reaction is "elementary", then it will be $a$th order with respect to $A$ and $b$th order with respect to $B$. This doesn't have to be true but often is. This means that if you change the concentration of A while holding everything else constant, the change in rate will be proportional to the concentration of $A$ to the $a$th power, etc. In math form that looks like
$$\dot{\xi} \propto [A]^a$$
$$\dot{\xi} \propto [B]^b$$
where $\propto$ means "proportional to", which is saying that the things are equal except for some one unknown constant. Thus those two $\propto$ "equations" are equivalent to
$$\dot{\xi} = \kappa_1 [A]^a$$
$$\dot{\xi} = \kappa_2 [B]^b$$
It's important to note that $\kappa_1$ could depend on everything except $[A]$, i.e. it could be a function of $[B]$ etc. Similarly $\kappa_2$ could depend on $[A]$ (but not on $[B]$). The only way for both equations above to be true is if
$$\dot{\xi} = k [A]^a [B]^b$$
This means that $\kappa_1 = (k [B]^b)$ and $\kappa_2 = (k [A]^a)$. Now that we know the reaction order with respect to each reactant, and have written it down in equation form, we can relate it back to the change in concentration of each of the reactants/products:
$$\frac{d[A]}{dt}=-a k [A]^a [B]^b$$
$$\frac{d[B]}{dt}=-b k [A]^a [B]^b$$
$$\frac{d[P]}{dt}= p k [A]^a [B]^b$$
$$\frac{d[Q]}{dt}= q k [A]^a [B]^b$$
These equations clearly show the relationship between the order of a reaction and the rate of change of reagent/product concentrations.
I have learnt for many years that rate of reaction is the rate of change of concentration of a reactant or product with respect to time.
By looking at the four equations above, the rate of reaction will only be equal to the product or reactant concentration change when the stoiciometric coefficient of that product or reactant is 1. And that's true regardless of what the "order" of the reaction is. The reaction $\ce{\it aA + bB-> pP + qQ}$ doesn't have to be $a$th order with respect to $a$. It could have been that $\dot{\xi} \propto [A]^{1.7}$ or even that $\dot{\xi} \propto \frac{[A]^{0.2}}{K_m+[A]}$ or whatever. But in chemistry, it does often happen that reactions do have orders that depend straightforwardly on stoichiometry, which is why those types of reactions are studied so often.
