Specific format/formula when changing column datatype
First and foremost: 


Why are you storing dates as text?


Second, as I pointed out in comments, there is an accepted answer of Aaron Bertrand that states: 'Sorry, not possible', you cannot change the default conversion behaviour. And that is why I flagged this question as duplicated.

And third, having a look at Microsoft SQL Server 2008 Internals, at chapter: Internals of Altering Tables. (bold is mine)


  Another negative side effect of altering tables happens when a
  column is altered to increase its length. In this case, the old
  column is not actually replaced. Rather, a new column is added to
  the table, and DBCC PAGE shows you that the old data is still there.
  I’ll let you explore the page dumps for this situation on your own,
  but we can see some of this unexpected behaviour by just looking at the
  column offsets using the column detail query that I showed you earlier
  in this chapter.
  
  Another drawback to the behaviour of SQL Server in not actually dropping the old column is that we are now more severely limited in the size of the row. The row size now includes the old column, which is no longer usable or visible (unless you use DBCC PAGE).   


You can check the length by using DATALENGTH function:

DECLARE @dt datetime = GETDATE();
DECLARE @sdt varchar(20) = CONVERT(varchar, GETDATE(), 20);

SELECT @dt AS dt, DATALENGTH(@dt) AS dt_lenght,
       @sdt AS sdt, DATALENGTH(@sdt) AS sdt_lenght
GO



dt                      | dt_length | sdt                 | sdt_length
:---------------------- | --------: | :------------------ | ---------:
2020-05-30 10:50:51.587 |         8 | 2020-05-30 10:50:51 |         19


db&lt;>fiddle here