You are right about different standards and regulations in different countries, so your previous experience from Russia might not entirely apply here.
Sorry to say this, but a site foreman is not a small position for a graduate, that`s someone with extensive and hands on field experience. You can check up on seek or trademe and see what qualities employers are looking for in a site foreman position.
Most local people here don`t actually care about degrees, they prefer experience more than anything. I personally think the higher education, the better, and a big builder would likely prefer a higher degree. It only puts employers off when you only have the high degree but no real world experience. In your case, a higher formal education diploma or degree is probably worth more considering you are aiming to get enough points for PR.
I only suggested national diploma in construction management level 6, because it seems to me that it`s a more popular choice, the programme gives more practical and extensive course options, and it has longer study period which would allow you more time to look for a job while on the study. A graduate diploma level 7 is still just a diploma, not a degree. Correct me if I am wrong, but it seems to me that people who choose to study level 7 are already working in the local construction industry and just want to advance their project management skills and knowledge. Level 7 has shorter study period, which is a factor you have to consider, will that give you enough time to look for a job? Do get a job in the field while are you attending school, otherwise CM is an expensive place to only learn the ropes. A bachelor`s degree of construction level 7 would have been ideal, but it requires longer learning period and more courses to go through, and I am not sure about their admission requirements.
As for what programme you should enroll, that all depends on what your goal is for the future, do you want to work for yourself as developer or work as a project manager at small residential construction firms, or you just want to look for a stable job at bigger construction companies that do commercial projects. And among commercial do you want to work for main contractor or sub contractor. I guess you`ll come to realize what you really want once you have lived here long enough and gained enough experience. The project managers for main contractors that I know of are almost exclusively kiwi. It seems cool to work for a big company, but you don`t necessarily make more money. On the other hand, jobs like that are normally more stable, have lower risk, and expose you to more opportunities to big projects (comparatively speaking, NZ big not worldwide big, lol). It`s possible to get a job with sub contractors, for example, how about a project manager for a steel fabricator? Given that you have previous experience working with metals and machines. Hopefully you`ll be lucky enough to meet an employer who will be willing to take you in. Or how about a diploma in quantity survey? But if you`ve got your mind set on construction management, just forget about what I said.
Anyway, good luck! Sorry If I couldn`t be of more help.