Four formats. The practical questions are whether it needs to stand on a shelf and whether someone writes inside it. Both of which are settled by stock rather than by design. Filter by format.

Center score, full interior, someone writes a note inside.

Flat, printed both sides, announcements and thank-you cards.

Cut to a silhouette. The shape is part of the message.

Square, panoramic or gatefold, no standard format fits the idea.
A folded card gives a full interior panel for a handwritten note and stands unaided on a shelf from about 300gsm. Those two things are why the format dominates. A flat card costs less, posts lighter and reads as more contemporary, but there is no inside, so the message shares the back and it will not stand without something to lean on. Choose by whether the card is displayed or filed.
Whatever the format, check two things. A gloss surface rejects pen entirely, so anything meant to be signed needs a matte or uncoated interior even if the outside is gloss. And coated card at these weights must be scored before folding, or the coating fractures white along the spine on the first fold. On shaped cards, settle the envelope before the die is cut. It is the constraint people forget.