no, they're both different styles of internal Northern Kung Fu. I don't know much about Ziranmen, but Bagua is based on moving/walking in circles (a common joke is that people practising Bagua forms look like tops ;p) - they fight from the outside of the circle (circumference) in toward the centre.

Xingyi, Taiji and Liuhe are also different forms of internal Northern Kung Fu. Xingyi is more linear than Bagua - taiji is basically the combative form of tai chi. That site doesn't mention which style of Taiji they teach (e.g.: Yang, Chen, Wu etc) so I can't tell you much more than that. I don't know much about Shaolin Liuhe, my knowledge of Six Harmony is mostly in regard to Liuhe Tanglangquan (Six Harmony Praying Mantis), and even there it's admittedly quite limited - all Northern Praying Mantis is classified as internal, and from what little I've seen of Six Harmony Mantis it's a traditionally "soft" internal mantis form.

Classes/lessons don't have individual names. You will have names of styles and the forms (set pattern of movements) in which they practice.