Based on FAQ Help and F.A.Q.
Postcrossing is a postcard exchange community, so each exchange that you make must include at least one postcard. It can be a postcard you bought in a store or a handmade postcard, as long as it hasn’t been previously used.
Is folded card a postcard? I don’t know. I think Postcrossing team should define what postcard is clearly in their rules or FAQ.
Well, stamp is more expensive than postcard, isn’t it?
A folded card in an envelope does cost more within the US and to mail international, I do not know about rates from other countries, but would assume it varies. Perhaps that is all those persons had to send at the time, I don’t know… but they could be gently reminded that it should not be a folded card but a postcard. And a postcard can be adapted from a folded card if not too thin. But 2 or 3 received is a bit odd I think
Similarly to what others have mentioned above, I would suggest to contact the admins for guidance rather than telling people on your profile not to send you folded cards. Some folks can be very sensitive to wording on profiles and if you take into account that many people do not speak English as a first language, there could be misinterpretation of your intent.
I understand the frustration of getting folded cards rather than postcards since it seems to defeat the purpose of this site, but in some cultures/languages, the vocabulary for both are the same. And considering the supply issues due to COVID, not everyone will have access to a variety of cards. It’s possible that they only have folded cards available. Or after reading your profile, they believe that the folded cards they have fit your interests better than any of the postcards they currently have on hand.
They may also be reluctant to transform the folded card into a more traditional postcard because I’ve seen people on this forum vehemently post that they don’t think cut up folded cards are in any way postcards.
Postcrossing is about kindness, so the kind thing to do is register the folded card (realizing that not all languages differentiate a folded card from a postcard), and in your thank-you note about how very nice it was to receive their card and how happy it made you, mention politely that they may have a better experience Postcrossing if they send postcards rather than folded cards because not everyone is as kind are you are.
In my profile, I would write that people should send me Genuine postcards. Folded cards are not postcards in my opinion. I would still register it. I learned that apart from real junk (bookmarks, cardboard cut-outs, etc). it’s not worth it to contact PostCrossing admins. They will tell you to register it.
We discussed a similar issue in the following topic: What if someone cuts the front of a folded card to send it as postcard? Then it becomes a “handmade postcard” which is fine by the rules…
I also recommend not to write what you wouldn’t register, because that sounds harsh in my opinion (and for me it’d take the fun out of writing a card if I got your address), but to express that you’re not a fan of folded cards as they’re not postcards.
Once I was called cheap because I was arguing that folded cards are not such a sin… The funny thing is that everywhere I have lived in my life, a folded card costs like 5 times more than a postcard… I definitely wouldn’t choose them for economic reasons!
Though maybe people have folded cards lying around and not postcards… I don’t know.
I agree that addressing it with the moderators is a better option than putting it in the profile. I understand it is not a postcard but it is true that saying some things won’t be registered is a bit too much.
The few times I received folded cards (maybe 2-3 times in total) I registered them, since I really don’t mind them. In at least one case I tried to gently explain the sender to be careful as a lot of people don’t consider them postcards. I click “register” and find out that member had thousands of cards sent so they should be aware! Anyway I left it at that.
Everything adds.
However, I agree with the opinion that folded cards use to be more expensive than postcards. But there is another issue there: in many places you can find folded cards but there are no postcards available. In my village there is just one place to buy postcards: a stationary shop with only four horrible multiviews from the town, while there are a handful of places where you can buy lots of different folded cards.
There was an answer by @mundoo regarding folded cards on the old forum: Postcrossing (Legacy) Forum - Should I register this?
The choice is the receivers but if it is registered we do suggest a polite message about the difference between the cards as usually it is a new member who has sent it.
In either case letting Postcrossing know (with scans of both sides) lets us contact the member about the issue.
The last time I received a folded card, I was very disappointed and let the card sit for two days to think about what I want to do with it. As there was a nice message I registered it then (telling the difference to a postcard). And I was happy about my decision as I noticed it was the first card of that user.
I think we have to register every card which has been sent in the official Postcrossing with the PC code in it. Also the pieces of paper or selfmade drawings by a 3-years-old child. You can tell in your profile that you like to receive only postcards, but some of the members have no idea what is the difference between postcards and folded cards. For example, in Finland, we call them all as “postikortti”.
Nasi-chan is new in this Postcrossing and starts to make it a hobby.
This “folded-card” is new thing for Nasi-chan, here people call it (literally translated) as “Greeting Card” (such as Christmas, Birthday, etc card).
Personally, Nasi-chan agree that folded card or greeting card is not a postcard.
But, Nasi-chan still think that he would register it if he received one.
Who knows, Nasi-chan has not receive his first card yet T-T
This is not an excuse because they can also not send a card.
Also I actually strongly dislike this image of poorer people being that selfish that they wouldn’t care about how other feel of what they send and they would overlook the rules.
Maybe you can write something about that you are here to swap postcards, not folded cards, and you won’t register folded cards at once, but will ask if you need to register folded cards.
I only advise to write this, if this is how it really was. Otherwise, when for example if I write that it wasn’t a postcard, they start complaining how everyone else has always loved it bla bla bla: and I should be thankful.
Postcrossing is about exchanging postcards, there is even a picture of a postcard where is address, postage and message on the same side. They should be able to handle a little correction if they hurried to join and not even read the f.a.q. It benefits everyone.
If I remember correctly, in some languages such as Dutch there isn’t a distinction between folded cards and postcards, they are called with the same name. I read it somewhere in the forum, and I think that may be the source of confusion.
Still, it’s better to contact the admins/mods when we’re not sure what to do.
Folded cards are no postcards. And the rules say that you have to send postcards.
I agree with “stating you won’t register sounds harsh”; you could try to phrase that better, maybe asking please not to send fold up cards (it’s difficult to try to explain something with the language barrier, but you could try something along that line…)
We all pass differently stages on Postcrossing, times we care for the postcard, front, back, times for the stamp…I’m actually at the stage “I’m happy with any piece of paper that arrives as long as I feel connected” 
No we don’t have to. If it’s not a postcard, I write to postcrossing team and they decide.
Maybe someone sometimes did need to register a drawing as this example seems to appear every now and then, but I don’t think it applies to every member every time.
Postikortti is the one without envelope.
Taittokortti, kaksiosainen kortti is used for folded card that is sent in an envelope.
And in general, if people don’t have an idea, here’s a chance to learn. I don’t think we are supposed to let them believe there is no difference in these, when there is.
Here I could buy folded cards cheaply in boxes or for example in a paper shop, I think 2 or 3 card with 1 Euro. And average price of postcard is about 1,60€. But here postage is the same. So sometimes the greeting cards are much cheaper.
But someone answered that postcard was “ansichkaart” and folded “venskaart” or something similar, if we are remembering the same conversation.
Oh, I didn’t remember that! Thanks for clarifying, it seems that my memory is not that reliable 
I think this has been discussed in so many topics and I can remember something from the old forum 
Maybe many just say “card” in whatever language, and don’t think it so thoroughly.
Well… The rules say ‘postcards’ - “All POSTCARDS must be registered” 
Folded note-cards are not postcards, are they? This has nothing to do with not matching one’s personal preferences - they are simply not postcards, so it is the sender who violates the rules in the first place, not the person who doesn’t want to register this “letter on a note card” that was sent instead of a postcard 