Does “trimmed for current speed” mean that the plane will do whatever to maintain the current airspeed?
The pitch trim setting is based on the angle of attack.  The angle of attack is affected by indicated airspeed, center of gravity and any unbalanced forces.  

In a steady state aircraft where the airplanes forces are all balanced then we are left with airspeed and center of gravity.  Under normal operations where fuel is not being transferred from an aft tank to the wing tanks or vice versa, center of gravity for a certain period of time remains relatively constant.  This leaves airspeed as the primary change in angle of attack.

Your analysis would be correct.  If you are trimmed for 290 KIAS (not KTAS like you stated) and you increase thrust the following sequence will occur.


Airplane will begin to accelerate
The lift formula comes into play and additional lift is created on the wings causing the airplane to pitch up and start to climb.
The airplane experiences an increase in induced drag which opposed thrust.
The airplane will now begin to decelerate while still climbing.
The decreased airspeed causes lift and induced drag to decrease.
The aircraft will pitch down and start to descend.
Aircraft will begin to accelerate as it descends.
The cycle continues until the phugoid oscillation dissipates (provided the airplane demonstrates positive dynamic stability)
In the end, the airplane will be at a higher pitch attitude (in relation to the horizon), climbing, and at 290 KIAS.


The same type of analysis can be accomplished if airspeed is reduced.  The airplane, at the end of the phugoid oscillation, will be at a lower pitch attitude and the same airspeed.