Is it the 6 year-old's responsibility to remember their coat before going to school, or is their parent's responsibility?
The answer will depend on a lot on all parties involved and your expectations:
Is your child able to remember to take the coat? And bring it back home?
I have two children that were both raised to be independent and responsible - Yet, my 9 yo still struggles to remember basic things (there is a reason he owns 4 pairs of gloves...) and we had days he would have left for school in his pajama shirt if I hadn't stopped him. He needs the occasional reminders. My 6 yo might ask which coat, but that's it.
Your parenting style:
Are you more of a "helicopter parent" or a "hands-off parent"?
I've met parents who brought their 16 yo the forgotten lunch parcel to school - although we had a perfectly fine cafeteria. Incidentally, this is the same mother who now "helps" said daughter (38, stay at home mom) to raise her child and run the household. My own approach was always to let the child try for himself first, only interfere when said child asks for help or to prevent serious accidents. So ask youself, where you will draw the line.
Your tolerance for failure and the real risks involved:
With our son, we've come to expect a certain "loss rate" and we plan and budget accordingly. And if he forgets his snack or his homework, he has to deal with the consequences. Honestly, what will happen? He won't starve until lunchtime and yes, he might have to do some extra homework - neither will kill him. As far as the coat in question: Will your child experience discomfort (cool day) or lasting effects (frosty day)? Then decide.
Consider where you and your child are in this spectrum, then decide. You know your child better than we do.
It's the parent's responsibility to set up the environment to encourage success. In the specific case of remembering to wear a coat on cold days this may mean, for example, keeping coats on hooks by the door instead of hung up in a closet with a door that's kept shut. It may mean saying "coat!" as the child opens the doorknob to leave. It may mean inventing a "ready to go" song that the child and parents sing before going out the door. (I put things I don't want to forget into my shoe the day before, or use magnets to affix a note that covers the doorknob so I can't leave without interacting with the note. There is nothing wrong with using such approaches to help a child do what needs to be done.)
Nobody would argue that a newborn should remember to put on their coat. And nobody would argue that a 35 year old who lives away from their parents should somehow blame their parents should they find themselves at work without a coat. So clearly there is a point in a child's life where the parent is totally responsible, and point where the child is totally responsible, and a point in between where they share that responsibility. If your question is, "is age 6 the point where it's totally the child's responsibility?" then the only possible answer is "that depends on the child, and the consequences for forgetting."
Where I live, it's coldest in the mornings when children leave for school (about 8:30 am) and warms up as the day goes on. If a child is cold when they first get out the door, they can loop back inside and grab the coat. If not, then they are likely to be fine all day without it. They may feel slightly uncomfortable for a few minutes, but that is harmless and will provide good motivation for bringing a coat next time. This may not apply where you live. Ask yourself the consequences of forgetting the coat. Will the child get frostbite or other serious damage? (I don't believe you get colds from being cold, but even if you did, a cold is not serious damage.) Will the child be forbidden to play outside at recess or lunch, and have to do something less pleasant? Will the child receive a punishment of some sort from the school that is not a logical consequence? Will the school demand that a parent deliver a coat?
If the consequences are mild, logical, and apply only to the child, I would probably still remind the child (making it their responsibility, but helping them with it.) If they ignored the reminder, I would let the consequences be theirs. If one day I forgot to remind them, and they later tried to say it was my fault the consequences occurred, I would refute that. "It's up to you to decide about your coat" I would say. I remember I had to send a written waiver with my 4 year old (which was kept on file) that said she could decide whether to put on her hat and scarf (and especially whether her scarf was to be tied over her face with just eyes peeping out) when coming home on the bus in the afternoons. She was drenched with sweat but the school's position was "you send a scarf, we make them wear it." (I sent a scarf at 8:30 am because it was cold then.) So at age 4 I definitely took the position a child can decide how much outerwear to wear for themselves.
If the consequences are more severe (physical damage) or you don't support them (school forbids library books to those who forget coats) or the consequences affect you too (have to go to the school to get the child or deliver a coat) then I would be less willing to let the child choose to accept those consequences. My reminders would probably be more forceful. Because we live in the country, and drive long distances in heated cars, my children didn't always want to wear their coats, boots, or snowpants. My rule was "you don't have to wear it, but you have to bring it" in case the car was to break down. This is a matter of life or death where I live. You don't have your coat and boots, I'm not starting the car. I would take the same approach with your 6-year old. It's up to you whether you wear it or not, but you have to take it with you and you have to bring it home.
If you find reminding tedious, or you keep having to go retrieve coats from the lost and found or the far end of the playground (been there!) you might consider imposing some logical consequences for that, such as extra chores to help make up for the time you spent dealing with the child's forgetfulness. But I would also work with the child on learning how to remember things so that they can fully shoulder this responsibility (which remains theirs; you're just helping) sooner rather than later.
It is the child's responsibility to remember their coat. As a parent you are supposed to be teaching them to become their own people, you won't be there to watch over them forever. If you do not give children responsibility, they will never learn it. It's a complicated subject and the sooner you start, the better.
However, it is the parent's responsibility to make the child does not come to harm. This means that if forgetting a coat means the child might get frostbite or undercooled or something like that, it's up to the parent to make sure the child still has the coat.
But this should not be seen as "taking responsibility for making sure the child brings a coat", it should be taking responsibility as a parent to protect the child from harm, and one of the steps of this protection should be reminding the child of their own responsibilities.
(On the other hand, if it'll just get chilly but not dangerous, letting them go without a coat might help them learn that bringing a coat is a good idea if it's cold out. This is also a way for them to learn responsibility.)
This is a tough one for me to answer to be honest; however, I will answer by saying it's up to the child.
My 8 year old Gabrielle is really forgetful at times, I am forever saying to her "Gabby, jacket please, its torrential rain outside" then it's "I said it's torrential rain so I think it'd be a good idea to zip your jacket up and put up the hood" before going out in morning.
She needs constant reminders at school and home which in my opinion is a worry but we get by though.