Difference between "response" and "reply"
Solution 1:
I would say that a reply is a type of response. But that not all responses are replies.
For example someone can have a response to a stimulus, such as a pin pick on their skin, which might be a reply
James pricked Carol with a pin. Carol slapped James, in reply.
The slap is also a response. However
James pricked Carol with a pin. Carol winced with pain.
The wince is a response, and not a reply.
A reply has to go back to the source of the stimulus of the response, e.g. a reply letter is written to someone who has written to you.
A response does not have to go back, e.g. someone who is not the questioner but is voting on an answer here is not replying, because the answer is not directed to them, but this is a response.
The second constraint is that if the stimulus for your response (or the response from your stimulus) did not come from an agent that you have anthropomorphised, then it is not a reply.
So you do not get replies from machines, but you can get replies from companies. You can get responses from both.
E.g. (thanks to Mr. Shiny and New)
I am moving my mouse but the pointer is not responding
not
*the pointer is not replying
also
I switched the key in the ignition, but got no response.
not
*I switched the key in the ignition, but got no reply.
But you can say
I wrote to MacDonalds about the quality of their beef, and they replied.
Solution 2:
"Reply" normally refers to words or some other form of communication. "Response" could be words or it could be actions. Thus I agree that "reply" is a subset of "response".
Right: I asked for his name, and he replied, "Bob".
Also right: I asked for his name, and he responded, "Bob".
Right: I pushed down on the gas pedal, and the car responded by surging forward.
Unlikely: I pushed down on the gas pedal, and the car replied by surging forward.
People occasionally use "reply" metaphorically. Like, "I asked his name, and he replied by punching me in the face." But this isn't a literal use. The sentence is deliberately constructed to give the idea that his response was a substitute for words in a situation where one would normally expect words to be used.