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In actual practice in general aviation, commercial aviation, etc, a climb is NORMALLY carried out a higher angle-of-attack and lift coefficient than we'd use for high-speed cruising flight. It's more efficient this way, and it also gives us the most climb performance out of a given, limited amount of thrust available. Why? Because a high lift coefficient also correlates with a high ratio of (lift coefficient to drag coefficient), which means a high ratio of lift to drag. For shallow to moderate climb angles, the higher the L/D ratio we can achieve, the steeper we can climb for a given amount of thrust. This is explored in more detail in the first link given in this answer. To look at climb rate rather than climb angle, we'd have to look at a chart of (power-available minus power-required) at various airspeeds or various angles-of-attack, but we'd come to a similar conclusion-- our best climb performance will be achieved at an angle-of=attack well above what we'll be using in high-speed cruising flight.

In actual practice in general aviation, commercial aviation, etc, a shallow to moderately steep climb is NORMALLY carried out a higher angle-of-attack and lift coefficient--and therefore a lower airspeed-- than we'd use for high-speed cruising flight. It's more efficient this way, and it also gives us the most climb performance out of a given, limited amount of thrust available. Why? Because a high lift coefficient also correlates with a high ratio of (lift coefficient to drag coefficient), which means a high ratio of lift to drag. For shallow to moderate climb angles, the higher the L/D ratio we can achieve, the steeper we can climb for a given amount of thrust. This is explored in more detail in the first link given in this answer. To look at climb rate rather than climb angle, we'd have to look at a chart of (power-available minus power-required) at various airspeeds or various angles-of-attack, but we'd come to a similar conclusion-- our best climb performance will be achieved at an angle-of=attack well above what we'll be using in high-speed cruising flight.

Your question included the statement "if I pitch the airplane up, but also increase power and am able to maintain the same speed, then no, the AoA hasn’t changed, although it may have varied in the transition between one situation and the other." For shallow to moderate climb angles, your statement is true for all practical purposes, but it is not EXACTLY true. If we want to be very precise about it, we could note that since lift is slightly reduced in the climb, if airspeed stayed constant than angle-of-attack must have been slightly reduced, and if angle-of-attack stayed exactly then airspeed must have been slightly reduced. This same idea came up in these two related answers to related questions, though in these cases the lift vector was reduced because the aircraft was in a descent rather than a climb -- 'Gravitational' power vs. engine power and Descending on a given glide slope (e.g. ILS) at a given airspeed-- is the size of the lift vector different in headwind versus tailwind?

In actual practice in general aviation, commercial aviation, etc, a climb is NORMALLY carried out a higher angle-of-attack and lift coefficient than we'd use for high-speed cruising flight. It's more efficient this way. A high lift coefficient also correlates with a high ratio of (lift coefficient to drag coefficient), which means a high ratio of lift to drag. The higher the L/D ratio we can achieve, the steeper we can climb for a given amount of thrust. This is explored in more detail in the first link given in this answer.

In actual practice in general aviation, commercial aviation, etc, a climb is NORMALLY carried out a higher angle-of-attack and lift coefficient than we'd use for high-speed cruising flight. It's more efficient this way, and it also gives us the most climb performance out of a given, limited amount of thrust available. Why? Because a high lift coefficient also correlates with a high ratio of (lift coefficient to drag coefficient), which means a high ratio of lift to drag. For shallow to moderate climb angles, the higher the L/D ratio we can achieve, the steeper we can climb for a given amount of thrust. This is explored in more detail in the first link given in this answer. To look at climb rate rather than climb angle, we'd have to look at a chart of (power-available minus power-required) at various airspeeds or various angles-of-attack, but we'd come to a similar conclusion-- our best climb performance will be achieved at an angle-of=attack well above what we'll be using in high-speed cruising flight.