Forests you control vs Forests in play
When cards refer to "forests," they are always referring to the land cards and not to the mana they produce; when referring to mana, they specifically say "mana" or use the appropriate mana symbol (including X where necessary to be abstract).  That means that in both cases, it is referring to the principle you are thinking about in your first statement.

For example, contrast the wording on Dungrove Elder with that on Omnath, Locus of Mana. One refers to "the number of Forests you control", and the other refers to "each green mana in your mana pool".

The difference between the phrasing "you control" and "in play" is the difference between controllers - in the former, it only counts forest cards on your "side" of the board, where as in the latter, it refers to all forests controlled by all players, totaled up.