The main way to improve at solving tactical puzzles - and at seeing tactical opportunities/threats in games - is to develop pattern recognition for the many tactical themes there are. Training on specific themes for some time is a very powerful way to do so. I'm experimenting with this approach on BraiMax Chess and results from users are very good - I'll publish some stats as soon as I get enough data.
Because of that I doubt doing problems of the former kind will help seeing forks much when going back to “find the best move” type problems.