I recently replaced a ceiling light fixture with a ceiling fan with two light sockets that have a maximum rating of 9 W for LED bulbs. The bulbs in the original fixture are 10 W LED and dimmable. The bulbs that came with the fan are non-dimmable. Can I use the 10 W bulbs in the 9 W max socket? My understanding is that heat is the main reason for the wattage rating, but does 1 W make that much of a difference? That’s barely 0.01 A if I’ve done my math correctly.
There are two completely different thermal objectives
First, don't set other stuff on fire. That's decided by the thermal insulation of the fixture (think: thick insulation in down lights) and limits the size (in actual wattage) of any bulb, but it's mainly aimed at incandescents. Incandescent bulbs love hot places and work better there.
A fixture succeeds at this goal if it has plenty of insulation and creates a little "oven" for the incandescent to burn in.
Second, let the LED keep cool. The LED cannot tolerate high temperature - neither at the LED proper or the electronic driver. Its worst worst enemy is itself: even with their good efficiency, still nearly 85% of its actual wattage is turning into heat at the LED or driver. The heat will cause premature failure of the LED.
A fixture succeeds at this goal if it provides an easy way for heat to be convected away from the bulb location.
These objectives are at best unrelated, and at worst conflicting. For instance a well-insulated, embedded-in-ceiling downlight might protect from incandescents but cook LEDs. An antiquey lamp with "lamp chimneys" might work great to convection-cool LEDs, but with incandescents, scorch the ceiling above it.
The fixture's LED limit applies to the second thermal objective. It is saying "Above 9 watts, we can't guarantee we can keep that LED cool enough for it to have a long life".
I think you misunderstood the instruction manual. Where it reads "LED bulbs (9-Watt, Maximum)" is in the parts list. This is not giving the fixture rating. It is saying that the included lamps will use a maximum of 9W. As the manufacturer may source the lamps from multiple vendors, this makes sense.
You will need to look at the fixture to determine the actual rating. There should be a sticker or stamping with the information.
The manual is vague, but I think the trick here is what I like to call "old wattage".
Back in the not-so-dark dark ages of light bulbs, everything was incandescent. Incandescent wattage was important because a light bulb with a 100 watt fillament puts out a LOT more heat than a 60 watt bulb. So if you put a higher watt bulb into many fixtures, it could do things like start fires. As such, you'll see a lot of fixtures rated for, say, "60 watt bulb maximum".
CFL and LED changed the rules because they produced the same lumens (measurement of light) without all that wasted heat. But people didn't learn lumens, they learned "old wattage". They want the same light that a 60 watt incandescent bulb produced. As such, CFL and LED are often labeled in "old wattage". I have a bathroom fixture that says "60 watts maximum", but I have a "75 watt" LED that uses a paltry 11 watts. It works because there's no serious heat generated.
That brings me to your light. It says "9 watt maximum", but that can't possibly be a limit. In order to have a hard 9 watt maximum, it would ridiculously small wiring (1 amp = 120 watts, and most small wires will carry that little easily). I seriously doubt your fixture has wires that small. But if we compare a 9 watt bulb to "old wattage", we find it's really a "60 watt" bulb.
Illuminating with the equivalent of a 60-Watt incandescent bulb but only consumes 9W of power, making these daylight led bulbs one of the best energy efficient lighting.
This now makes sense, because a lot of fixtures (especially fan lights) have labels that say "60 watt maximum". If this were a hard 9 watts, there would be warning labels screaming that fact everywhere.
It's already puzzling out of the box: the manual reads "uses two 9.5-Watt LED bulbs which are included", and the sockets are rated "maximum 9W".
Considering the horizontal position of the bulbs and that the light diffuser bowl is top-open and allows convection, it seems that 10W could be fine.
How the legal line would be drawn if there is a catastrophic failure at such a small margin of difference remains unclear. Strictly, 9W is the maximum, unless the manufacturer can clear you. Call them.
Whether an LED bulb will be suitable for use in a fixture will generally depend upon two factors: (1) how hot the bulb would get in the fixture, and (2) how hot the bulb could get without adversely affecting lifetime. If a fixture has a 9W rating, that would suggest that a typical 9W bulb may get warm enough to reduce its lifetime somewhat, but not disastrously; the reduction of lifetime would become increasingly severe if power levels go much beyond that, but predicting how far beyond one can go may be affected by many factors and thus hard to predict. A 10W bulb which is never operated at an ambient temperature of over 30C may have its lifetime affected by heat less than would a 9W bulb used in the same fixture when the ambient temperature was 45C.
If one has some 10W bulbs and a 9W fixture, and would have no use for the bulbs other than the fixture, the 10W bulbs would almost certainly work safely for however long they last. The lifetime might be reduced enough to make the purchase of 10W bulbs for such a purpose uneconomical compared with 9W bulbs, but if one has the 10W bulbs and no other use for them, and if one doesn't mind the effort required to replace bulbs when they fail, use of the 10W bulbs shouldn't pose any safety hazard.