This is in large part caused by two things that are different than most countries. One is that there's less of a stigma in the Scandinavian countries against one whom commits suicide or their family, including more support, including financial, for families affected by suicide.
Secondly, the Scandinavian countries are not at all leery about calling some deaths "suicide", such as maybe a person taking a hair-pin turn in their Volvo at high speed going off a cliff. Also, families can still collect on their life insurance policies-- at least the last time I've heard
Thirdly, the long darkness of winter pays a toll as well, and even the personalities of Swedes tends to change from summer to winter. Know as being relatively quiet and introverted people, this is far less true during the summer months.
I remember quite a few years ago running across a comparative study that found that U.S. suicide rates are almost as high as those in Sweden if we here in the States compiled stats in the same way the Swedes do. However, I don't know if this still would be true.
It's an interesting point. The OP didn't link to any stats, but rather a 2011 story which used Scandinavian countries as an example without stating any hard figures.
It's at least partially based on different reporting methods used to determine when a death is a suicide (as you have indicated). Kind of a legacy myth from the 60s and 70s that Scandanavians kill themselves at a particularly high rate. Myth might be too strong a word, I guess, but still...
In terms of 2015 WHO figures, Nordic countries rank as follows;
Finland - 35
Sweden - 46
Norway - 102
Denmark -105
Meanwhile, the top 5 countries are Sri Lanka, Guyana, Mongolia, Kazakhstan and Cote de Ivorie.
I suspect the linking of happy countries to high suicide rates is probably overstated, and quite possible completely misrepresented by the Time article mentioned in the OP.
List of sovereign states by suicide rate - Wikipedia