Nash equilibrium two players in a joint effortful project
You are actually almost correct. Your first approach shows the normal form of the game, but in an incomplete way, which is why you find "extra" Nash equilibria. The possible strategies of the players include all positive effort levels not just the ones you show.

The NE (0.25;0.25) is not actually an equilibrium because either player can reduce their effort a tiny amount (but less than the amount in your game) increasing their payoff. For example the optimum amount of effort for player 1 if player 2 uses effort level 0.25 is 0.1875 which is not in your game. If player 1 opts for an effort level of 0.1875 player 2's optimal effort is 3/4*0.1875=0.140625.
Continuing this reasoning you'll find eventually there is only one NE: (0;0) which is what you had correctly identified using your second approach.