When I inspected the dimension style being used by that dimension, I still found this setting (labeled 'Aligned Dimension: Inline - Hosizontal' when you hover over it) set to 'horizontal'...

...so I changed that to the one called 'Aligned Dimension: Inline - Aligned'...

...and that changed the dimension text to be aligned with the dimension line...at least at first.
But then, if you drag it around a bit, or run your rule on it, which simply moves the dimension test, it then wanted to form a horizontal line with the dimension text at the end of it (now horizontal text). I'm honestly not really sure if that is avoidable. I think you just have to make sure the text placement is either between the dimension lines (and not too far above or below where it would normally be), or directly off to one side, while still being relatively in-line with where it would normally be. The settings on the 'Display' tab of the dimension style editor dialog is where you can dictate where you want everything to be by default. There are also iLogic rules out here on the forums about re-aligning, and re-centering all kinds of dimensions according to these outlines after the fact. This little detail is just one of the many reasons that full, custom drawing automation is so complicated to achieve.
Wesley Crihfield

(Not an Autodesk Employee)