So, we have some strong opinions in here. That's fine.

Some points still to be made?

ALWAYS consider the manufacturer's break-in recommendations! If something goes wrong and you weld it into one expensive paperweight, you might consider blaming the mfr, and go to them for "satisfaction." (Some comments in other threads suggest that some of us DO think that way.) A mfr will be able to tell, and fairly easily, if the engine has been brutally abused, but nowadays would probably cave to prevent "bad press."

I also check that a new engine has been properly assembled... You'd be surprised at some of the mass-production goofs I've caught before first run... And, again, I've seen occasional comments from people unhappy with an expensive engine when they found a head bolt or backplate bolt hadn't been properly tightened. Well, duhh, didn't you even LOOK at the engine before operating it?

Proper torquing is important, too. You'd be surprised if you loosened all head bolts, checked how free the engine feels that way. Carefully torque the head back in with a consistent sequence, checking that you haven't introduced any bind, after each 'rotation' through the sequence. Modern engines are more substantial than those of the 1940's and 50's, but they can still suffer from bad torquing - consider: brass is not as hard as steel and most sleeves today are based on brass...

Type of engine makes a difference - and the mfr generally suggests what they are confident will work reliably. Examples: Some engines have synthetic oil only recommended; some recommend 0% nitro fuels; some say bolt it in and go fly... The differences I'm familiar with after almost 60 years of using glow, diesel and occasional spark engines seem to work just about every time:

Lapped iron pistons in steel sleeves - bench break-in, outdoors. Significant castor oil % in the fuel. Short runs - just up to thoroughly warm - and good cool-down before the next noise. Slobbering rich for first one or two runs, then gradually leaner over the next several. (First, polish in the rod bearings. Then go for the longer term heat-cycling and burnishing-in without galling anything.) At first sign of anything wrong developing, it is much easier to yank the fuel line off a bench-mounted engine...

Ringed engines are almost the same, except that the critical wear zone on the piston is only the height of the ring(s). If rod and shaft wear in properly, that small ring land shouldn't need as much run time to get durably happy...

Non-ferrous engines (Aluminum/Brass/Chrome, Aluminum/Aluminum/Chrome. Aluminum/Brass/Nickel), imho, still need at least one brief rich run to help the rod bearings run-in. THEN do as most mfrs suggest: several brief runs just below 2-cycling or warbling in/out of a rich 2-cycle. Cool-down between runs. DON'T wind in the needle (HS needle with RC carbs) until RPM sags! Sagging RPM can indicate heat expansion is pushing fits toward metal-metal contact. We have oil in the fuel to prevent that...

Why the difference? As Dar and others cited, modern CNC techniques allow cutting basic dimensions so that proper clearances only occur when operating temperatures expand the piston and sleeve the designed amount. Overheating these, when new particularly, can force metal-metal contact, which will gall finished surfaces, or wear away the designed dimensional fits. Loss of performance and power may not be a lot, but that's no way to help an engine work well for you for a useful length of time.

Engine temperatures: Engines reach about the same running temperatures pretty much regardless of ambient temps. The heat comes from burning the fuel. The complete cooling envelope limits the top temps reached. Our engines are partially air-cooled. Other cooling processes include vaporizing the fuel through the venturii or carb. Methanol chills significantly on evaporation, and the crankcase directly under the carb or venturii feels cool even on an engine running pretty hard. Do not try this on an engine buried in the nose of a model! However, it is safely accessible on a bench. The chilled zone adds a nice heat sink that doesn't warm from running. Another process is heating the oil going through the engine. Most of it doesn't add much heat to the combustion, even if some synthetic does burn away. Castor hardly ever burns, so it leaves the engine with the heat it has picked up. ...Physically carries those calories somewhere else... (Castor can char, and coat the carefully generated dimensions on the sleeve and piston, so it isn't always best as the only oil for non-ferrous, CNC produced engines. It can be good in a blend with synthetic oils, which act as detergents to reduce charring and buildup.)

A decade or two ago I was "breaking-in" numerous OS mid-size engines for an acquaintance's project that needed unquestionably settled-in ready-to-go engines. As many have commented, and occasionally the instructions will suggest, power and economy improve dramatically over the first 5 to 10 minutes of proper running. The first runs made for difficult starting, hot running and considerable dark matter flushed out in the exhaust oils. Economy was also pathetic... As these were modern, non-ferrous CNC made engines of well-deserved good repute, I understood the initial runs accomplished much burnishing of remaining machining tool marks and surfaces. (The machining was close, but not perfect.) Economy, handling and power rapidly improved as the oil cleared. BTW, I almost always hand-start engines on a test bench. It's a better "feel" for the engine's condition. Still have all my fingers, and only an acceptable scattering of small scars. ...Cautionary reminders to be careful, that I never got from an electric starter...

...Engines continued to improve over some additional running. The rate of improvement declines as you get it settled in. No sense just burning fuel, gassing neighbors and listening to that sweet music any longer than the engine basically needs.

This has also been the norm for several of my own OS and other non-ferrous modern engines. At least they reach a usable plateau much sooner than the iron-in-steel setups, as those metals do not have the higher expansion rates that allow cutting-in designed tapered fits.

For RC Carb engines, it makes more sense to set idle and transition AFTER the initial wear, as heat, handling and economy all shift as the engine approaches its break-in plateau. Mfr instructions often indicate that you should either wait until after a half-hour or hour of running time to achieve final carb settings, or at least recheck them after a similar runtime total.

The flying I most enjoy is CL Precision Aerobatics, where absolute engine reliability and consistency are crucial. I still like the sound of the 4/2 break style run, but recognize that there are more of the modern engines NOT designed to do that. Several of my flying buddies try to force the newer 'greyhounds' to 'mush a dogsled...' Recent engines can be controlled more easily by operating them more like what they were designed for. For CLPA, flatter pitch props and more RPM restrain excess speed, AND fit the non-ferrous engine design package better. The older engines operated nearer torque peak - around 9000 RPM - and weren't revved to anywhere near max power. With flat pitch props, A** engines are at higher power, but mixture limited, RPM, loafing along with that power on call if needed for maneuver loads...