Learn English – it called when words are deliberately spelled incorrectly but pronunciation is kept unchanged

Based on the example you've given, I think the most clear answer is:

Advertisement and Marketing.

Words like "nite" as in "Nick at Nite" or "thru" as in "Drive Thru", "tonite" as in "Tonite Only", even "donut" as in "Dunkin' Donuts", are all marketing and advertisement inventions--mostly of the American variety. While donut predates Dunkin' Donuts by about 100 years, most Americans who spell it without the -gh are purchasers of Dunkin' and other chains that have adopted the abbreviated spelling.

I remember reading in an Advertising textbook that the spelling choices in Nick at Nite were based on a desire for balance in the design of the logo: two four-letter words separated by "at", and the removal of the descender "g" to keep a clear visual line and give the two words a squared-off appearance:

I suppose there's a similar mentality in removing the descender "g" in thru, tonite, lite, et al.

The Chicago Tribune published this in 2011 about misspellings in advertisements:

"There's a lot of trickery involved in the way grammar is used in advertising," says Stephen Hahn-Griffiths, Chief Strategy Officer at Leo Burnett ad agency. "You're trying to differentiate yourself. You're trying to be part of the lexicon. You're trying to be part of popular culture."

And in the minds of consumers, Hahn-Griffiths says, flawed spelling doesn't equal a shoddy product.

"Look at La-Z-Boy," he says. "Everybody knows that's not the way you spell lazy, but it's one of the most powerful furniture companies there is. It's an attempt to connect with the vernacular and the way people speak."

A misspelled name works, he says, if it does one or more of the following:

• Helps with recall. "You want people to remember your name in an easy and anecdotal way. It's this cognitive dissonance thing that makes you remember it because it's just weird."

• Sets you apart. "Maybe you're doing something with your name to make people think you're better or different than other hair dressers, garages, cafés."

• Gets people talking. "At the end of the day you want to become part of the lexicon. It gets people talking about you, so you become part of the vernacular."

•Works as an onomatopoeia. "There's some kind of double entendre association based on the misspelling." (Onomatopoeias are those words that imitate the sound they represent: buzz, kerplunk, etc.)

Clearly, the effect of such misspellings is the stimulation that comes from the visual interruption of seeing common words in altered forms, which sometimes is doubly-impactful when uncommon letters like "x" as in xtrem sports and "z" as in Cheez Whiz are substituted. This stimulation triggers memory, which in turn triggers brand loyalty.

To give this type of misspelling a little more context, there were efforts in the early 20th century to simplify the spelling of English in the States. Andrew Carnegie founded the Simplified Spelling Board, which sought to simplify and shorten the spelling of many common words so children could memorize them more easily, and so typesetters could save a few cents spelling "wished" with a -t instead of an -ed, wisht. The New York Times, The Chicago Tribue, and Melvin Dewey of the Dewey Decimal System were all supporters of SSB. Before them, Benjamin Franklin and Noah Webster also supported spelling reform on the basis of simplification and phoneticization. With such strong support for spelling reform, it makes sense that the taboo surrounding misspelled words in big print media outlets lifted.

While the SSB disbanded because Carnegie believed that its spelling suggestions were too prescriptive, the rationale carried on in the scholarship of Advertisement and Marketing where manipulated spelling, as in the title of this book on advertising: "Spelling Manipulation and Present Day Advertising" (1923), was a common trope of the field. Today the reasons to misspell words in adverts go beyond logo designs. With the importance of google ad words, for instance, a company that can brand themselves with a misspelled name can purchase ad words on the cheap and can likely find open web addresses with greater ease.

So, while there are terms like manipulated spelling, which advertising scholars have used for nearly a century, and there are any number of these somewhat facetious terms:

Faux-urban misspelling: Using "z" in place of "s," "da" instead of "the," and "dogg" or "dawg" instead of "dog."

Xtreme misspelling: Using "x" is place of "ex," as in "Xtreme" or "Xtra."

Country misspelling: Using "k" in place of "c," as in "kountry" or "korner."

it is better to understand some of the motives of advertisement and marketing to give a more foundational understanding for why these spelling choices are made.