Builtin `Begin` not adding variable in context and `End` leaves traces of variables defined by `Begin[]`

[rocky]

Description

Begin is not adding context, and End might not be removing variables added.

How to Reproduce and Output Given

In[1]:= ??f
Out[1]= Global`f

In[2]:= Begin["MyContext`"]
Out[2]= "MyContext`"

In[3]:= f[x_] := x^2

In[4]:= ??f
Out[4]= Global`f  (* Wrong. Should be MyContext` *)       
        f[x_] = x^2

In[5]:= End[]
Out[5]= "MyContext`"

In[6]:= ??f
Out[6]= Global`f (* Wrong. Should have been removed *)
        
        f[MyContext`x_] = MyContext`x^2

Additional context

Noticed while trying to understand why certain Mathematica packages are failing.