Shortening words hasn't been about saving SMS credit for a decade. It's a way to communicate in casual, less formal tone. Without vocal flair you have to get creative to communicate emotions through text. Shortening words is just another tool in your arsenal, just like capitalization, emojis or literal tone indicators (like /s). It shows that you are very comfortable talking to someone when you are willing to forgo proper grammar and spelling. Just like we do in real conversations.
The younger generations have had online communication figured out since forever while older generations continue to be perplexed by it.
At least two decades! I'd say this trend goes back to the late 90s -- think chatspeak on AOL chatrooms, ICQ, etc. I didn't get my first mobile 'til '02, but SMS was invented in '92. People were probably doing this as soon as SMS was available on their phones.