という

という at a sentence's end belongs to written narration, citing no source (明治めいじの頃ころは宿場町しゅくばまちだったという). そうだ is the everyday spoken choice, and usually has a source, named with によると.

The same と sits over a finished plain form. 言いう lands in someone's mouth, 思おもう in the speaker's own head.

と言いう packages a name or idea as content, open-ended after it. と言いえば instead launches a fresh topic, by whatever the speaker connects to N.

という states the content as settled, often with no speaker in view (値上ねあげするという話はなし). と言いっていた reports one person speaking once, so it takes a time word (さっき言いっていた).

という can sit under any naming or content noun and stays open-ended (話はなし・うわさ・意味いみ). ということだ narrows it to こと, and with だ reads as a report or a conclusion.

という names something for a following noun, and needs one right after it (「はな」という店みせ). というのは stops there instead, making the word itself the topic.

って is the casual spoken contraction of という／とのことだ, freely dropping だ (来くるって). と言いう is the neutral choice, equally at home in writing, and never drops だ.

ドアを開あけると言いった quotes what he said (he'd open it). ドアを開あけると寒さむい states a condition instead, and only the quote と relaxes to って.

という hands on a quote with no feeling in it (辞やめるという話はなし). なんて always carries a verdict, delivered right after it.

と言いった is the plain past of と言いう, "said." といった lists members of a category (犬いぬや猫ねこといった動物どうぶつ).

Kanji 言いう reports words someone actually spoke aloud. Kana という just names or packages content, with no speech behind it (「はな」という店みせ).

という names content with nobody speaking or listening in view (値上ねあげするという話はなし). と言いわれる keeps both roles alive, said to the one reporting it.