Throughout the history of our awesome CMS, figuring out how the word is capitalised hasn't always been easy. The confusion is understandable. Over the past 10 years, the content management system has seen different ways of capitalising the name and there have been various logos to go with it.
Officially though, the name is all-caps: MODX. This is answered on the official MODX site in the form of a Frequently Asked Question:
I believe the relaunch and rebranding the FAQ refers to, is the one that happened around the CMS Expo, back in May 2012. There, the logo went from having many stripes in a ton of different colours, to just three simple, coloured stripes forming an X. Even more recently, for MODX its 10th birthday, the logo was simplified some more, bringing us to the folded logo that heads up MODX today. As expected, in full caps.
So why is there so much confusion?
Well, a search on Google Images for MODX Logo may give an idea.
Over the years, the project has seen many logo variations and they haven't always used the full uppercase name. Especially the old logo from the Evolution days (which was eventually discontinued because of striking similarities to another logo) shows why some of the older community members may still prefer the lowercase "modx". "MODx" has also been incredibly popular for a couple of years and many tutorials and blog posts around the web still use that particular capitalisation.
There is also another reason people use other variations: Code.
And then there's the PHP core code, which uses modX as primary class, and again, that's not something that can be easily changed.
So now you know! It is most definitely MODX when talking about the product, but when talking about the core code, MODx or modX is just fine.