The 4L60 & 700R4 are the same thing. The 1993-later 4L60E, however is not. The 'E' is for electronic and uses the aforementioned solenoids for shifting as opposed to a governor and TV cable.
Since your transmission is not simply hanging in second gear as evidenced by your comment about going to a neutral state, it's not the 2-3 shift valvetrain in the valve body or governor as they are still functioning. Sorry to say, but I too conclude that your 3-4 clutches are history. They aren't necessarily burned up. With over 200K miles (esp. if the transmission has never been rebuilt), they're probably just worn too thin to get the job done.