I use the word zen to refer to the stories and conversations of the zen characters as found in the cases, from the case anthologies like the BCR, Gateless Gate, etc. So, its a limited use of the word zen.
For chan, I accept that chan is considered a part of Chinese Buddhism. And since the zen characters I am interested in don't seem to make it a priority to pin their talk of seeing and freedom to any particular place, and especially since most of the power of chan came to bear after they died, I do not find the use of the word chan as meaningful in the study of the zen stories and conversations. And the history of chan, especially after Yuanwu is not of much interest to me, and this is where most of chan's history lies. Before Yuanwu, it is more fruitful to focus on other categories, such as
1)The period of the 6 patriarchs
2)The Tang period zen characters
3)The song period zen characters
4)The song period Song Buddhist orthodoxy
5)The period in China from 150 CE to Bodhidharma when Buddha teachings were introduced
6)The period in India before Buddha ideas were introduced to China
But, still, I think it is safe to say that there were chan elements in the time of the 6 patriarchs that did not make it into the the field of interest of the zen characters of the Tang period.
Some of what we say about the zen conversations and stories that we think happened with the zen characters during the Tang will be shown to have been invented later, in the Song period. A lot of what we think happened in the Tang actually did.
People often have a hard time with the zen stories and conversations from the Tang; these stories evade normal conceptual grasping. Impatient, people then look to interpretations, or they look to earlier and later periods to second guess what the zen characters of the Tang period were saying or doing.
I don't think this second guessing has worked. But that means a lot of people will scratch their heads, or get impatient. Oh well. People are going to do what they are going to do. There is no solution to that. It just means that if you are not all that interested, you are going to get distracted, or follow other interests. Nothing wrong with that. But then, calling that zen, or chan, or zen buddhism, whatever, some might disagree.