Interesting question. It's a pity that the first answer option is ambiguous, with one reading making it correct and the other incorrect, while the second option, which is claimed to be the correct answer, is just plain incorrect.
The definition is changed to use the new name in the text defining the proc's name; that's one interpretation of the first option, and is the only correct answer.
The definition is changed to use the new name everywhere it had the old name; that's another interpretation of the first option, and is an incorrect answer.
"The definition is untouched" - anything that says that is wrong, because the definition of a stored procedure includes its name, which is changed. It's in the first bit of the text returned by SELECT object_definition(Object_id(<name of sp>)) when that returns any text, rather than NULL. The explanation mentions that stored procedure, so it's a bit surprising that the question's author apparently didn't notice it. So the second option is incorrect. Of course the second option also states that the procedure now contains the new name, which is true because the body contains that name, but since the second option also makes the incorrect claim that the definition is untouched the second option is not a correct answer.
Some of the earlier comments suggest that renaming SPs (and views and so on) is more of a problem that renaming other things like tables. I don't agree. Renaming most things is fraught: that's because everything that uses the name has to be altered. Tables are no easier than stored procedures: every script or stored procedure that uses the table has to be changed to use the new name. So does every view and every function that uses the table, and since views and functions can't call stored procedures that makes a table more awkward to rename than a stored procedure. This always makes me think that people who claim rename is less of a problem with tables than with SPs are not actually correct. I guess it depends on the extent to which sys.object_definition() and the sys.sql_modules view are important; to me they don't seem to be very important at all - SSMS gets the SP definitions correct when asked to script them and the sql system invokes the correct SP when asked to despite object_definition(), and hence sys.sql_modules, delivering NULL for the definitions of renamed SPs (and SSMS gets it right views and so on, too). When I want an SP or View definition, I ask SSMS for a script. When I want to call an SP or use a view in a DML statement object_definition() isn't involved. So why should I care that object_definition() can't provide the definition of a renamed view or trigger or SP or whatever? Oddly enough I do care, because maybe object_definition() will in future be needed for something I want to handle; so I shouldn't use sp_rename where it breaks object_definition(); but equally I can see that because object_definition doesn't work for some renamed things, in the future maybe it won't work for others - so I shouldn't use sp_rename for tables either. In practice the only things I have ever used sp_rename for in production code are databases (but I use it for prettty well anything when not in production).