How Input and Output Work
|Input||convert from a textual form to an expression|
|Processing||do computations on the expression|
|Output||convert the resulting expression to textual form|
When you type something like x^2 what the Wolfram Language at first sees is just the string of characters x, ^, 2. But with the usual way that the Wolfram Language is set up, it immediately knows to convert this string of characters into the expression Power[x,2].
Then, after whatever processing is possible has been done, the Wolfram Language takes the expression Power[x,2] and converts it into some kind of textual representation for output.
It is important to understand that in a typical Wolfram Language session In[n] and Out[n] record only the underlying expressions that are processed, not the textual representations that happen to be used for their input or output.
If you explicitly request a particular kind of output, say by using TraditionalForm[expr], then what you get will be labeled with Out[n]//TraditionalForm. This indicates that what you are seeing is expr//TraditionalForm, even though the value of Out[n] itself is just expr.
The Wolfram Language also allows you to specify globally that you want output to be displayed in a particular form. And if you do this, then the form will no longer be indicated explicitly in the label for each line. But it is still the case that In[n] and Out[n] will record only underlying expressions, not the textual representations used for their input and output.