Q: When was the art named “Taiji” boxing 太極拳?

The name Taiji, or Tai Chi, was used already before Yang Luchan. It was used in ‘Taijiquan Lun’ written by Li Helin in 1787 (we know this from the Li Family Manual).
Whether the name was used in the Chen family is unclear, the Chen Family had only a few original Tai Chi texts in their possession around 1900.
The Yang and Wu Yuxiang lineage could have known the name  of the art from the classic ‘Taijiquan Lun’ that was in their possession, obtained from their acquaintance of the salt store in Wuyang owned by the Li Family.
However, the name Taijiquan could, in fact, also have been transmitted to Yang Luchan from the Chen family. We just can’t tell from written sources. After Chen Wangting’s time we have no sources to tell us if the Li family members who wrote the classics had any further connection with the Chen family. But the Li Family were, after all, distant family at some point, so there is a real possibility for this.