This claim is very misleading. There are a total of 5 deaths directly linked to the January 6th, according to my research. NYT link
I found it is misleading with the claim specifically highlighting "On January 6th" since there were deaths linked with the storming of the capital.
To highlight the main claim, that there was only one death on January 6th, I investigated obituaries and/or causes of deaths related to January 6th.
Obituary #1 clearly states Ashli Babbitt, who is mentioned in the New York Times article above, died January 6th, 2021, in Washington, D.C.
NYT about Kevin D. Greeson, although I could not identify a secure obituary for Mr. Greeson, there are tons of articles, similar to this one, that support the claim that he suffered a heart attack amongst the rioters.
The MSNBC podcast states that Rosanne Boyland died on January 6, 2021.
Obituary #2 This is the obituary for Benjamin Philips, who died on January 6th, 2021, after suffering a stroke within the chaos.
This is already 4 people who died on January 6th, not to mention the deaths of others after complications stemming from January 6th or suicides related to the incident, including Brian Sicknick who was attacked by the mob and died of complications on the 7th of January, 2021, Jefferey Smith who killed himself after the event, and Howard Liebengood who also took his life afterwards. NYT LINK
This claim's wording is very misleading. Because, yes, these other deaths were incidents that don't include someone killing them, yet they still died on January 6th, or later.