If it's the 9 year old, it makes more sense.
We had the same sort of problems when difficult child 3 was younger. We would give time warnings but we would be prepared to moderate it a little, along the lines of, "How long do you think you need, to get to finish, or a save point?" We would then give them t hat time limit (usually set within reason, once they realise you are going to let them get to a logical stop point). if the time they gave was way too far in the distance, we would say, "You know the time to switch off is 8.30 pm, if you know the game is going to take you way past that time then you shouldn't start it."
But a bit of leniency does relieve the anxiety that they won't be permitted to finish the game.
Witz, we don't have "you can play for an hour," instead we have "you can play until such-and-such a time," but we also rank games according to what time of day they may be played. Less need to now, but we've had a few games which we had to ban difficult child 3 from for a while, because they got him too worked up, or gave him nightmares, especially if he played them in his evening games bracket (after 7 pm).
We never had computer games set to grades, because we stopped homework so early. But when difficult child 1 was still doing homework (and later, schoolwork at home) I did require a certain quantity of work to be completed at a certain standard before I gave him back the missing parts (RF modulator, usually). This was a restriction he had agreed to himself - we negotiated.
We have found that negotiating these arrangements with them, as much as possible, has helped them realise that our aim is for them to do well and be OK. But we keep our negotiation for times when they are calm and medication is fully on board. Last thing at night is not the time.
What we would do in your situation - exactly what you did. BUT next morning, we would have sat down and said, "Last night you weren't coping too well. You were getting really anxious about a fictional cat, at a time in the evening when your brain needs to be winding down for sleep. Now, we're not saying that you may never play that game again, but we do think that perhaps you need to stop playing that particular game by ...[name a time you think is suitable]".
Be open to negotiation. Listen to her. She is likely to insist it's not a problem for her at all, and want a later time. You can reply with, "Honey, we care about you and don't want you to have trouble disengaging. And you did last night, so that time was clearly too late. How about we just try this for a few nights? What other game can you play after that, that won't get you so worked up? There are some really good brain training games, and some good puzzle ones where you're not racing the clock. Which ones do you think would be good to let you play a game but still help you settle down for bed?"
Trial it. If she really insists hard on being permitted to play the cat game, then there are two things you can do -
1) Talk the game through with her. How can she more effectively satisfy the cat, without getting so stressed? Is it a no-win situation, or is it something she CAN do when she gets better at it? Nintendogs is a doggy example of puppies being dissatisfied until you meet their needs, but you aren't set up for failure with it, you can do it fairly easily. difficult child 3 was actually encouraged to play with his Nintendogs game, by his psychologist, who said it was really good for his social skills and empathy development.
2) Tell her you will give her three nights' trial with playing the cat game at the later hour. You do not think it is a good idea but you want to give her a chance to try to control herself. However, if she cannot do it, she will have to accept your alternative from then on, for another month (or negotiable time). When she is older and more able to control her stress in response to computer games, then she can play it later. And if she can show she can handle it earlier in the day, she may continue to play it at other times providing she is getting her other chores done, of course.
By involving them in the decisions, they are learning self-control and judgement, often at a younger age than they would be asked to.
We found that our boys do seem to be addicted to games, but by being open about it and setting certain PRACTICAL rules that they help establish, we get honesty and cooperation.
When difficult child 3 would get nightmares from a particular game, we had to have rules that he was not to be in the same room as anyone else playing that game. As he got older, he would briefly watch and seemed OK, so we taught him that if he was playing it and finding it stressful, he could always pause it and walk away.
He now plays that game without any problems. It's a fairly primitive game which doesn't progress until you choose what action to take - a D&D type of thing called Mission Thunderbolt. It's actually taught him to take control of his anxiety and to not panic - if his character is in crisis, he simply stops making ANY move and the game automatically pauses while he considers the options.
difficult child 3 is less anxious because he knows we are not merely setting unfair rules arbitrarily. He knows he can be part of the discussion and decisions, as long as he is also reasonable. And a ban as punishment is usually something he's agreed on ahead of time. Usually though, we don't ban games as punishment because we see them as part of his coping strategies. I think it's the same for a lot of difficult children - they use games to burn off excess mental energy as well as to escape. But in our house, they also have to get their own tasks completed. Games must stop at 7 pm so the evening routine can be followed - chores, bath, dinner, bed. If these things are completed before 8.30 pm, he may go back to gaming until then. After that, he may read or with special dispensation play a gentle puzzle game or educational game. He may also play in the morning before school, as long as he gets everything else done and stops at 8.30 am to do half an hour maths revision online before school.
When we stopped difficult child 1 from gaming because in his case it really WAS interfering with getting schoolwork done, I used a similar tactic - I asked him to set the number of hours a day he could play. Naturally he set what he thought was an outrageously generous time, but I knew that he had no clue how much time he was gaming. So I let him have his way - and within a couple of days he was climbing the walls with gamer withdrawal and agreeing that he was playing way too much, to have underestimated like that! He was 16 at the time, old enough to be reasoned with and to be motivated to do well at school.
I hope this helps.
Marg