I have a Master of Science degree from the EECS department at MIT. From the website it is pretty clear they call it an SM degree. I think many US universities refer to a Master of Science degree as an MS degree. My UK department has offered for a long time a program that leads to a Master of Science degree that we call the MSc program, but next year we are introducing an MSci degree. Do the different abbreviations officially mean something? Is it dishonest to refer to my SM as an MSc in the UK? If not, is it helpful?
While all these degrees fall under the category of Master of Science there are technical differences in the course content. In general you should always refer to your degree by its actual title/abbreviation. If you think someone might not know what an abbreviation is e.g. on your CV or something, you could add a brief note such as MS (US Master's) but I'm not sure how useful this really is. Google has all this information readily available.
For the particular degrees you list:
I've no idea if there is a difference between an SM and MS, but I'm not in the US.
A US MS is generally two years compared to the UK MSc's one. I also the impression that the MS is more research focused, although that might just be my view of things.
An MSci is a 4 year combined bachelors and masters that is increasingly popular in the UK. Generally students only do a single research project/dissertation in their 4th year which is slightly shorter than an equivalent project for MSc students. So having an Msci is not exactly equivalent to an MSc.
This is before you get to some of the more obscure things such as MRes which some UK students do. These are still broadly at Masters level but have different focuses to the course.
Just to be clear, I would consider it somewhat dishonest to translated the abbreviation of your degree.
There is no difference: M.S., M.Sc., and S.M. all mean Master of Science. The difference for S.M. is that it is in Latin: scientiae magister.
I have no idea whether it will actually help anybody's confusion to translate back to MS, but there is certainly no question of honesty. For anybody with a Ph.D., however, I expect it will not make the least shred of difference, as a Ph.D. supersedes it quite effectively.
Harvard University uses SM for its Master of Science degrees. Please see https://www.seas.harvard.edu/news/2017/06/harvard-launches-data-science-masters-degree-program.