According to this site, they operate with the following terms around a minimal monthly insurance fee:
- Beitragssatz - Mindestbeitrag EUR 1038.33
- Krankenversicherung - 163,02
What is now the mandatory monthly payment to have the insurance?
The terminology on that site is a bit confusing. Where you think they are talking about the Beitrag they are actually talking about the Bemessungsgrundlage.
The Bemessungsgrundlage is your monthly income from which the fees are calculated. There is a maximum at around 4500€ beyond which income is ignored. For full-time self-employed people there is also a lower bound for the Bemessungsgrundlage, which since this year is about 1000€. If you make less money than this, your fees are not reduced beyond the fees at the minimum Bemessungsgrundlage.
You can then calculate your Beitrag (monthly contributions) by multiplying your Bemessungsgrundlage with the Beitragssatz, which is a percentage of your income which you have to pay. Note that there are separate percentages for Krankenversicherung (health insurance, here: 15.1–15.7%) and Pflegeversicherung (care insurance, here: 3.05–3.3%) which are legally distinct but must be payed together.
As per the tables in the page you linked, the minimum payments with Barmer are 156.79€ for health plus 31.67€ for care, a bit more if you don't have children or if you want Krankengeld.
All the Gesetzliche Krankenversicherungen (public insurance) use the same mechanism and have the same Bemessungsgrundlage limits. However, they can differ a bit in the percentages, the Beitragssatz. You can also opt out of the public system and use private insurance, which can sometimes be cheaper as they may use flat rates instead of a Bemessungsgrundlage × Beitragssatz. But leaving the public system and its predictable rates is a mostly irreversible choice for life, so don't do that without understanding the implications.