symmetry - Transformation of $d^4x$ under translation disregarded?

Under a translation in spacetime i.e.,

$$x\mapsto x^\prime=x+a,\tag{a}$$ a scalar field $\phi(x)$ $$\phi(x)\mapsto\phi^\prime(x)=\phi(x-a).\tag{b}$$ My aim is to verify the invariance of an action of the form $S[\phi(x)]=\int d^4x~ \phi^2(x)$. This video by M. Luty shows the invariance (around time 17.40 minuties) as follows. $$\int d^4x~\phi^{2}(x)\mapsto\int d^4x~\phi^{\prime 2}(x)\tag{1}$$ $$=\int d^4x~\phi^{2}(x-a)\tag{2}.$$ Now by changing variables $x-a=y$, one has $$S[\phi(x)]\mapsto\int d^4y~\phi^2(y)=S[\phi(y)]\tag{3}$$ which readily establishes the invariance of the action.

Question I'm a bit confused about step (1). Since the coordinates also change shouldn't we also change $d^4x\to d^4x^\prime$ in step (1)? I know that differentials don't change by adding a constant to a variable. In fact, that's what is used in arriving at step (3) from step (2). But the step (1) looks like $\phi(x)$ is mapped to $\phi^\prime(x-a)$ and $x$ is mapped to $x$. I'm suspicious whether it is $d^4x$ or really $d^4x^\prime$ (which is in turn equal to $d^4x$) in step (1).

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$S_2[\phi]$ of AFT's answer Using (a) and (b), (when both the intergrand and $dx$ are changed) we get,

$$S_2[\phi]=\int x^2 \phi(x)^2 dx\mapsto \int (x+a)^2\phi(x-a)d(x+a)=\int (x+a)^2\phi(x-a)dx.$$ Changing variable $y=x-a$, I find $$S_2[\phi]\mapsto \int(y+2a)^2\phi^2(y)dy\neq S_2[\phi].$$ So it shows that even when $dx$ is changed to $dx^\prime$, the action is not translationally invariant.

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References

1. 
   

   A Modern Introduction to Quanrum Field Theory Eq. 3.19, 3.20 and 3.21.

   
   


2. 
   

   Field Quantization-W. Greiner Eq. 2.38, 2.39, and 2.45.

   
   


3. 
   

   An introduction to Quantum Field theory- Peskin and Schroeder page 18.

   
   


4. 
   
   

   Lectures on Classical Field Theory by Suresh Govindrajan

   
   


5. 
   

   Lectures on Quantum Field Theory by Ashok Das page 212, Eq. 6.4

   
   


6. 
   

   Relativistic Quantum Physics-Tommy Ohlsson Eq. 5.66, page 119.

   
   


7. 
   

   A first book on quantum field theory by P. B. Pal Page 22, Eq. 2.38.