As I stated in the comments section:
Actually, there is no right or wrong way to use the term -- or rather, the preferred term would depend on who you were talking to.After writing that, I did a little search, and I ran across an interesting website -- Britain 1906 - 1918 Contrast, Contradiction, and Change. As you have probably already figured out, even before clicking on the link, it's all about movements in England. Nonetheless, there are certainly enough similarities between the British Women's Suffrage movement and the American Women's Suffrage Movement to get an idea of what was going on. There are some great documents included on this site -- cartoons, photographs, correspondance, and articles. On the particular topic of Suffragists vs. Suffragettes, you can see, perhaps more clearly, the lines drawn between them.
The term Suffragette was coined by the London newspaper, the Daily News in 1906 -- scathingly they referred to the women as not real suffragists. By adding the "ette" diminuitive, it tried to ridicule the women as something small, almost like an imitation of the real thing such as one would compare a kitchenette to a real kitchen.
After that, many British suffragists, and a few American ones, adopted the term as a way to differentiate themselves from the staid constitutionalists who sought political equality through negotiation and lobbying. Most American suffragists, however, continued to use the term suffragist, choosing to not reclaim the insulting term.
At some point, suffragist came to mean someone who was fighting for the vote for women in a "peaceful" way, while the radicals (or militants), who would break windows, set fires, and go to jail were known as suffragettes.