The "official procedure" is the one dictated in your aircraft's POH/AFM.
Within the leeway that document gives you (and many of them give you a lot of leeway) this is definitely one of those "endless debate" subjects.
Personally in a single-engine piston aircraft unless the POH directs otherwise or operational considerations (like an altitude restriction from ATC) require something different I would maintain full power (and all other takeoff settings - fuel pumps, etc.) until reaching 1000-1500 feet AGL (depending on what the terrain and obstacles in the area are like), then configure the aircraft for a reduced-power cruise climb if appropriate.
Whether or not the first power reduction is a time of increased failure is irrelevant to me - I operate on the assumption that the engine can fail at any time, and should the engine fail (for whatever reason) I want adequate distance between myself and the ground to make some decisions, confirm my landing spot, and prepare myself, my passengers, and the aircraft for whatever is going to happen next.
1000 feet AGL is my minimum threshold for that sort of activity. It's possible (maybe even likely) that I'm imposing additional wear on the engine in requiring full power for this long, but if the engine decides to fail on climb-out it's not going to matter how "nice" I was being to it on all my previous flights - the only things that are going to matter are my altitude, airspeed, and landing options, and I'm going to want as much altitude and airspeed as I can get to help maximize my landing options!
There are some conditions where I would reduce power sooner than 1000 feet AGL if necessary - for example, if I can't maintain oil or cylinder head temperatures "in the green" in a full power climb a power reduction (or pitch reduction) to keep those engine parameters happy would be in order.
There are also some aircraft/engines which have a continuous power restriction (
Full throttle operation limited to 3 minutes,
maximum continuous power 2400 RPM, or similar admonitions you'd find in your POH). In those cases the restrictions should obviously be observed (the engineers who set the limitations had a good reason for doing so), and observing those requirements may necessitate a power reduction or other configuration changes prior to reaching 1000 feet AGL.
Finally, I'm not sure what kind of engine is in your hypothetical Cessna but as an additional point of reference Lycoming has this to say about takeoff operations:
Operating Your Lycoming Engines At Takeoff And Climb
Because there are
a wide variety of Lycoming engines in operation, the paragraphs below
may be helpful in understanding the different modes of operation
required when operating each type at takeoff and climb power settings.
The Pilots Operating Handbook for the aircraft in which the engines
are installed should be the final authority as to how the engine
should be operated.
DIRECT DRIVE ENGINES
Most normally aspirated
engines are rated at full power for takeoff and climb indefinitely,
provided engine temperatures and pressures are within the green arc
area of the engine instruments. Extra fuel, sensible airspeed, and
cowl flaps, if available, are all helpful in keeping cylinder head
temperatures within desired limits during takeoff or climb.
requirements may vary; as an example, on a warm day with the airplane
close to gross weight, and a direct drive engine with a fixed pitch
prop, the pilot will need full throttle all the way to cruise
altitude. The same airplane on a cold day and lightly loaded may not
require full power for climb. After full throttle at takeoff, the
pilot may want to reduce power 100 or 200 RPM and still not see
performance suffer. Those direct drive normally aspirated engines with
a prop governor are also rated indefinitely at full power, and the
manuals all recommend full power for takeoff, but specify a small
reduction in power, generally to 85% power climb. Study the specific
airplane Pilot's Operating Handbook for detailed power settings.
(They go on to talk about geared engines and turbocharged/supercharged engines, but your question was about normally-aspirated engines. This is from Lycoming Flyer "Key Reprints" - I grabbed it here but you can Google it up pretty easily, and I imagine Continental has analogous guidance somewhere.)