Doubt about request for nenkin payment in the mail

I live in Kanagawa, and today I just received a mail from the pension service containing two payslips to pay for National Health Insurance: one of about 17k yen referring to May of this year and one for April to October of last year totaling almost 120k yen. The former I was kind of expecting due to having received a mail before from them due to not being covered for five days back in the start of June – a period during which I was switching jobs, and therefore did not have shakai hoken for the first five days of June.

However, the latter is a bit puzzling to me: I did not have an employment visa in Japan from April to October of last year. During that time I was doing unpaid technical training at a research institute in Tsukuba (Ibaraki prefecture), and later got paid by them for a part-time (baito) contract from June to September, from which they automatically deducted ~20% of my salary as is common for foreigners during their first year in Japan as I was not considered a resident. The company itself told me to not apply for NHI, because at the time I was expected to go back to France by the end of September. It ended up not happening: during October I was unemployed and in the middle of changing visa status to Researcher, which happened around the middle of the month, and then started a new job at the same institute in November. From there on all my taxes and contributions were automatically deducted.

So my questions are:

  1. Is it the case that I needed to pay for NHI for that period of six months in 2024, or this is some kind of improper charge and I should contact some office to try to get it waived?
  2. Does this mean I am late on payments and they are charging me now? I need to know for sure as I intend to apply for PR in the future, and I know there is that rule about two years of no late payments (would the one payment in June of this year also count as a late payment)?

by Prestigious-Place941