arriving via plane from USA, departing via train to Mongolia

You must have a visa to take the train to Mongolia. Otherwise you will be refused boarding at your origin and will not get as far as Beijing. Visa-free transit does not apply to you.

Saul,
Thank you so much for your reply. I appreciate your information, but I am confused because the airline sent a confirmation of the ticket with the attached stipulations:
"Nationals of USA holding confirmed onward air, cruise or train tickets to a third country, arriving in and departing from any one of the following locations: Beijing (PEK), Tianjin (TSN), Shijiazhuang (SJW), Beijing West Railway Station, Tianjin International Cruise Home Port or Qinhuangdao Sea Port for a maximum of 144 hours, starting from 00:01 on the day following the day of entry. They must have a passport valid for a minimum of 3 months from the arrival date."
The Beijing West Railway Station is listed as a port of entry/exit both in the above text and on the Chinese Gov't web tool to determine eligibility for the Visa-Free Transit:
https://app.www.gov.cn/govdata/html5/2018/visafreetransit/pc/index.html
What information are you using to make your statement that a regular visa is necessary to take the train to Mongolia? I only ask, because it seems there are several publicized rules about this visa and it seems like you are aware of one that I want to know about! any information details you can give me would be greatly appreciated!!! thank you so much, -Erin

The train to Mongolia doesn't leave from the West Railway Station.
It leaves from the old main station and this is not a point of exit for the 144h twov.
Just the West Railway station is eligible for the 144h twov but no trains to Mongolia leave from there.

Thank you Thomas. That was most helpful.