Hi Stevenson,
Just a couple of quick points: if you want me to notice, use my handle @sujato. I don’t click on everything!
Also, please try to use the proper Pali diacriticals: several of the terms in your list are not displaying correctly. If that’s a problem, just use the plain letters.
The first list is, as you say, the classes of rules in the Patimokkha.
The second group is not really a group; they are various offences mentioned, but they are not organized. Pācittiya is a mistake there, it belongs in the first group.
Generally speaking, thullaccaya offenses are subsidiary offences for someone who has completed most, but not all, of the factors of a pārājika or saṅghādisesa. However some thullaccaya rules are also independent, not connected to pārājika or saṅghādisesa.
Similarly, dukkaṭa offenses are for partial offenses under pācittiya, but also for incomplete commission of a thullaccaya. And there are many independent dukkaṭas as well.
Dubbhāsita applies only to one kind of offence of wrong speech.
Where they are mentioned in order, presumably of severity, pārājika or saṅghādisesa are obviously most severe, then we find thullaccayaṃ, pācittiyaṃ, pāṭidesanīyaṃ, dukkaṭaṃ, dubbhāsitaṃ (Pvr 13).
Sekhiya is a slightly odd category, as breaking one is not in itself an offence. It just says they should be done. However it is usually interpreted that if you deliberately break a sekhiya out of disrespect, it is a dukkaṭa.
Adhikaraṇasamatha is another odd category, in that it is a set of procedures to be followed, rather than rules to be kept. Despite the fact that it appears at the end, it embodies many of the most important principles of the Vinaya.
Apart from pārājika, which entails expulsion, and saṅghādisesa, which entails suspension, all these offences may be cleared with a simple confession.