I spent a great deal of my undergraduate honors thesis dealing with this topic, and I think I can shed some light on it for you. I am trained as an art historian, and I focus on Japanese material culture with a specific focus on the evolution and symbolic role of weapons. I can confidently say that the Wikipedia article is largely incorrect.

The Mongol invasions affected Japanese swordmaking in more than one way. The immediate effect of the Mongol invasions on Japanese swordmaking included some minor adjustments to the way blades were made to make them more effective. The transition from the tachi to katana/wakizashi pair was a far more gradual process, which appears to have been in part started by the Mongol invasions. The Wikipedia article seems to be conflating these two effects.

The Metropolitan Museum of Art did a fantastic exhibition on Japanese swords in 2009, and you can access the catalog for free here. Starting on page 117, Harada Kazutoshi offers an excellent summary of the Japanese sword's history of development.

Now, on to your question. First, it is important to keep in mind that a Samurai's primary weapon was the Japanese longbow, or yumi. The focus of Samurai combat before the Mongol invasions was on mounted archery. Swords were used as a secondary weapon after two armies met. Samurai typically carried longer swords called tachi. They would also occasionally wield spear like weapons called naginata.

It is generally accepted that the Mongol invasions are at least part of what prompted dramatic changes in Japanese combat between the Kamakura and Edo periods. Foremost among these changes was a shift in emphasis from mounted combat to foot combat, which is what ultimately caused the sword to be shortened into the katana we know today. It is commonly held that after the Mongol invasions, warfare in Japan became less polite and warriors less skillful. However, exactly how that transition played out is still a matter of debate.

In 1986, Myra Shackley published paper in World Archaeology which detailed her forensic evaluation of remains from the Zaimokuza battlefield cemetery.^1 The battle in question took place in 1333, half a century after the last Mongol invasions. Shackley concluded that injuries on the remains were probably inflicted by swords wielded by mounted soldiers. She acknowledges that the swords could have been either tachi or katana. Given the preference for tachi for mounted combat, if we accept Shackely's conclusions then tachi are the most likely culprits for most of the wounds, indicating that their use was still widespread at least half a century after the invasion. Her final conclusion insists that pre-Mongol levels of skill and etiquette were alive and well in 1333.

However, in 2004 Shackley's conclusions were disputed by Antony Karasulas.^2 Karasulas argued that Shackley had made several errors in her paper, stating:

"I argue that Shackley has misinterpreted the damage on the Kamakura remains, and that the evidence actually indicates a very different battlefield scenario than the one she proposes. Her paper also contains factual errors concerning feudal Japanese sword use and terminology, and she makes an implicit assumption that the sword was the principal weapon of the fourteenth century. (emphasis mine)

Karasulas argued that the majority of the remains appear to be inflicted by Naginata, wielded by warriors on horseback. Karasulas found no evidence to indicate that any level of "exceptional sword skill or of remarkable battlefield ethics" are indicated by the remains at Zaimokuza.

Regardless of how the shift in warfare played out, according to Kazutoshi the switch to the Uchigatana did not happen until the middle of the 15th century. It first became popular among low-class samurai, with high class samurai following suit. The katana/wakizashi pair did not become standard until the last half of the 16th century. The Wikipedia article states that the Katana was invented in the 13th and 14th centuries, but according to every source I know, it did not become widespread until the late 16th century. There is far too much of a gap between the Mongol invasion and the invention of the katana for the invasions to be credited as the immediate reason for its invention.

Sources:

1: Shackley, Myra. “Arms and the Men; 14th Century Japanese Swordsmanship Illustrated by Skeletons from Zaimokuza, near Kamakura, Japan.” World Archaeology 18, no. 2 (October 1, 1986): 247–54.

2: Antony Karasulas (2004) Zaimokuza reconsidered: The forensic evidence, and classical Japanese swordsmanship, World Archaeology, 36:4, 507-518

See also:

Nagayama, Kokan. The Connoisseurs Book of Japanese Swords. First Edition. Tokyo; New York: Kodansha USA, 1998. Print.

Roach, Colin M., and Nicklaus Suino. Japanese Swords: Cultural Icons of a Nation; The History, Metallurgy and Iconography of the Samurai Sword. Paperback with DVD edition. Tuttle Publishing, 2014. Print.

Yoshihara, Yoshindo, Leon Kapp, and Hiroko Kapp. The Art of the Japanese Sword: The Craft of Swordmaking and Its Appreciation. Hardcover with Jacket edition. Tuttle Publishing, 2012. Print.

Feel free to ask me for additional references, I have an extensive bibliography on this topic.

Edit: Minor spelling errors.

Edit 2: I almost forgot. Part of the confusion results from the fact that samurai often passed down swords as family heirlooms. It was common for samurai of later eras to have their tachi shortened into katana. So it is common to sometimes see swords from earlier eras labeled as katanas. In reality they began as tachi, and were modified later.

Edit 3: My first gold ever! Thanks, kind internet stranger!

As a side question, did the Mongols actually wear leather armor? Most historians and reenactors I've heard on the subject have said that leather armor (at least in Europe) is mainly a myth propagated by D&D and other RPGs, with gambesons and brigandines often being mistook for leather armor. Was it different elsewhere in the world?

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