Translation:I will start running in a month.
"next month I will start running" was not accepted. Is there another way of saying next month in Chinese besides 一个月后?
Actually, next month can be a day away depending on what day of the month it it. In a month means around 30 days from now. Next month would be 下个月。
跑步 (Pao bu) is jogging; shouldn't "I will start jogging in a month" be accepted?
Maybe. I'm sure I've seen 跑步 defined to cover both "run" and "jog" although there's also 慢跑 which specifically means "jog". It would be good to hear from a native Chinese speaker.
Jogging and running are the same. In fact, jogging is a more accurate translation.