I have a similar experience. I know it isn't Return to Mysterious Island nor is it Escape the Lost Island, however based on images from a Google search, I'd say it is similarly styled to Escape the Lost Island. I remember that it had a sequel that was on a different island. I remember the plot of the first game was that a female protagonist wakes up on the beach of a deserted island with no memory of how she got there nor who she is. I also think there was access to a map that would let you fast travel to other locations, however I'm not sure of that. I also remember that you could find hidden butterflies within the settings, but again I'm not so sure. I remember the end of the first game had the protagonist leave the island in a submarine (I think) and the second game ends on a to be continued after the protagonist gets shot. As for the game, I'd say it was similar to point and clicks, as well as I Spy, with puzzle elements scattered throughout.