How to dynamically define module methods based on the class' instance variable?

You can't really do this without actually interacting with the module. When a module is included its included in the class which places the module on the ancestors chain of the class and fires the Module#included callback on the module.

Since this happens on the class level there is no way to access the context of an instance unless you're actually calling an instance method. In which case self is the instance of the class that includes the module:

module BaseFoo
  def extend_base_foo
    with_context.each do |context|
      BaseFoo.module_eval do
        define_method context do
          'lorem ipsum'
        end
      end
    end
  end
end

class Foo
  include BaseFoo

  attr_reader :with_context

  def initialize(with_context: %i[bar baz])
    @with_context = with_context
    extend_base_foo
  end

  def call
    puts bar # Should return 'lorem ipsum'
    puts baz # Should return 'lorem ipsum'
  end
end

But honestly why on earth would you ever want to do this? Creating an instance of Foo will actually define those methods on the module itself which means that they are included in every instance of Foo:

Foo.new(with_context: [:x, :y])
Foo.new.x # loren ipsum - big WTF moment

If you really want to do something like this create an anonymous module and extend the instance with it:

class Foo
  def initialize(with_context: %i[bar baz])
    m = Module.new do
      with_context.each do |context|
        define_method(context) do
          'lorem ipsum'
        end
      end
    end
    self.extend(m)
  end

  def call
    puts bar # Should return 'lorem ipsum'
    puts baz # Should return 'lorem ipsum'
  end
end