Itâs hard to evaluate exactly what youâre encountering from reading a textual description (that is a universal truth), but Iâll give it a shot.
Youâve established that an increase in power leads to a nose-up tendency and vice-versa for reducing power. But do you know why? Itâs because of the change in airflow over the horizontal stabilizer and how that interacts with the centers of lift (which might also be changing) and center of gravity.
If you were to balance an aircraft on its center of gravity (CG) at zero indicated airspeed, itâd just sit there. Now letâs start moving the aircraft - an increase of airspeed over an airfoil like a wing, when combined with angle of attack, results in an increase in lift. The center of that lift (CL), in most non ultra-maneuverable aircraft (like a Cessna), is coming from a point on the wing that is behind the center of gravity. So adding lift will naturally want to torque the aircraft nose-down ad-infinitum. If a continuous forward tumble were possible, thatâs what it would do.
Enter the horizontal stabilizer. The horizontal stab is way out there on the empennage, so it has a pretty good lever arm from both the CG and CL. However, it is also an airfoil. The key is that it âfliesâ upside down - providing lift, when accelerated, increasingly downward (as opposed to the wing, which provides lift upward). This tail-down force counteracts the aforementioned tumbling moment provided by the CG and CL, providing balance.
But the horizontal stabilizer isnât completely fixed in place. Otherwise, it would only provide balance at one set of airspeeds and weight/balance conditions, and thatâs not a thing in flying. So we can adjust it either through moving the trailing edge of it, as in an elevator, or the entire thing, as with a stabilator. So in flying the horizontal stab, you are grossly adjusting that tail-down force to maintain the pitch attitude you want.
But again, an increase in lift not only affects the wing, but it increases the tail down force of the horizontal stab. So you speed up, lift increases, the tail goes down, the nose goes up and you start losing airspeed (energy), which causes the lift to decrease, the tail to go less down (or up), the nose to go less up (or down), which causes the airspeed to increase, etc. donât perform any dampening, or worse, amplify them by means of pilot-induced oscillation (PIO) and the result is the phugoid oscillation you mentioned.
So in essence what we want to do is use the elevator to find a speed at which the acceleration of the airplane is stable - not increasing or decreasing airspeed, and a steady rate of climb or descent. Let me rephrase: when you pitch the aircraft, youâre pitching for speed. If I want to maintain 80 knots in a Cessna 172 at full throttle, I put the nose up until it gets to 80 knots and hold that speed using the elevator. If holding the elevator in a position to make it maintain that speed is tiresome (and it will be), I use trim to reduce the pressure, in essence balancing the elevatorâs angle of attack and resulting lift at the intended speed.
In that scenario, I take whatever rate of climb/descent I can get. But letâs say itâs giving us a 1000fpm climb and I only want 500fpm. In most realistic cases, weâd pitch for a higher airspeed and resulting lower climb rate, but no, in this example I want to maintain 80 knots and 500 FPM. I can do this one of two ways (without reconfiguring the airfoils) - one, I can take off heavier, or two, I can reduce the power of the engine. After all, itâs excess power or thrust that allows us to climb.
So I throttle back, and now the airspeed drops a bit, so I need to pitch down a bit and the airspeed stabilizes or picks back up until Iâve hit my target airspeed again and desired rate of climb. So the answer is BOTH - youâre going to be playing with pitch and power to achieve the desired performance, whether itâs takeoff or landing, itâs just that takeoff weâre most often at max continuous power no matter what so we take what we get. But landing is where it gets more⊠playful, usually due to the increase in drag that accompanies slower flight with flaps and gear out (another topic).
But as a pilot, a big portion of our job is to anticipate and dampen those oscillations weâre speaking of. So when you reduce power and do nothing else, youâre going to get behind the aircraft - itâll get slow, maybe to the point where you lose elevator effectiveness, or worse, exceed the critical angle of attack and stall. Instead, pitch for the desired airspeed and trim to reduce pressure. Adjust the throttle to get the rate or angle you want, then repurchase and retrieve to maintain it all. Practice, practice, practice and youâll get the hang of this.
To this extent, I wouldnât practice in the pattern - too much (notional) traffic, too many distractions, and too low to the ground. Instead, isolate the variables. Go out to a good practice area, several thousand feet above the ground. Establish trimmed, straight and level flight, then reduce the throttle. Aim for a particular airspeed and rate of descent. Use the combination of pitch and power to achieve a stable descent. When you achieve that, trim, trim, trim until you donât have to hold the yoke anymore.
Do the same for climbs. Try both climbs and descents clean and dirty (flaps and/or gear), see the differences. There is an order to doing this effectively, but I donât care about that in this moment - I want you to experiment. Try to anticipate how the nose and airspeed are going to respond to your changes in power and dampen them, then trim to hold them. The runway aimpoints and the transition to a landing attitude is another discussion that comes later. Practice and master stabilized descents and climbs first. 
As far as PAPI/VASI lights, I have a whole rant on how the game places then incorrectly, often way too far down short runways, to the point where I often ignore them unless I know theyâre placed correctly.