Why does this asymmetry exist?
Backward compatibility, and because the relationship between copying and moving is already asymmetrical. The definition of MoveConstructible is a special case of CopyConstructible, meaning that all CopyConstructible types are also MoveConstructible types. That's true because a copy constructor taking a reference-to-const will handle rvalues as well as lvalues.
A copyable type can be initialized from rvalues without a move constructor (it just might not be as efficient as it could be with a move constructor).
A copy constructor can also used to perform a "move" in implicitly-defined move constructors of derived classes when moving the base sub-object.
So a copy constructor can be seen as a "degenerate move constructor", so if a type has a copy constructor it doesn't strictly need a move constructor, it is already MoveConstructible, so simply not declaring the move constructor is acceptable.
The opposite is not true, a movable type is not necessarily copyable, e.g. move-only types. In those cases, making the copy constructor and assignment deleted provides better diagnostics than just not declaring them and getting errors about binding lvalues to rvalue references.
Why not just specify that if a move operation is declared, no copy operations will be declared?
Better diagnostics and more explicit semantics. "Defined as deleted" is the C++11 way to clearly say "this operation is not allowed", rather than just happening to be omitted by mistake or missing for some other reason.
The special case of "not declared" for move constructors and move assignment operators is unusual and is special because of the asymmetry described above, but special cases are usually best kept for a few narrow cases (it's worth noting here that "not declared" can also apply to the default constructor).
Also worth noting is that one of the paragraphs you refer to, [class.copy] p7, says (emphasis mine):
If the class definition does not explicitly declare a copy constructor, one is declared implicitly. If the class definition declares a move constructor or move assignment operator, the implicitly declared copy constructor is defined as deleted; otherwise, it is defined as defaulted (8.4). The latter case is deprecated if the class has a user-declared copy assignment operator or a user-declared destructor.
"The latter case" refers to the "otherwise, it is defined as defaulted" part. Paragraph 18 has similar wording for the copy assignment operator.
So the intention of the committee is that in some future version of C++ other types of special member function will also cause the copy constructor and copy assignment operator to be deleted. The reasoning is that if your class needs a user-defined destructor then the implicitly-defined copy behaviour is probably not going to do the right thing. That change hasn't been made for C++11 or C++14 for backward compatibility reasons, but the idea is that in some future version to prevent the copy constructor and copy assignment operator being deleted you will need to declare them explicitly and define them as defaulted.
So deleting copy constructors if they might not do the right thing is the general case, and "not declared" is a special case just for move constructors, because the copy constructor can provide the degenerate move anyway.