As I can't recall the event precisely and have no video of it,...
No, but you have those photographs, which clearly show that your car crossed the white line whilst red was showing.
I am not sure if I was slowing down the car and that's why I was moving forward.
Whatever you were doing, the photos show that you did not stop before reaching the white line. So you have committed the offence.
What exactly is the "rest of the story" which you believe the photographs do not show and which you think the court should hear?
I am obviously going to say I was the driver so I need to respond with this info,...
No you don't. Your partner does. You will then get your own notice asking you to confirm you were driving.
There were/are no signs for a red light camera showing or to my knowledge at this location, or is this not a requirement?
No it is not a requirement.
can you postpone doing the educational course...
If you are offered a course you will have to complete it within about four months from the date of the offence (the offer will provide the exact details).
...can you do it online? ie. can you ask for any special dispensations?
Most courses are held online. It will cost you about £100 which you will have to pay in one payment before you take the course.
My driving licence is at my old address and not my current one - should I get all letters sent to that address or change my driving address today?
Your partner should provide your current address when responding to the request for driver's details. You should do likewise when you receive your own request. It would be a good idea to change your D/L details.