Electronic – Please review the RGB Controller

It seems "possible to likely" that the relative polarity of the two controllers is reversed.
Whether the controllers and strips will work together depends on how they the strips are wired.

If you have the +ve and -ve ends of 3 colours of LEDs available (R+, R-, G+, G-, B+. B-) then you can probably just " turn the LED strips around end for end". BUT it is more likely that

• One controller expects common Anode LEDs where the +ve Anodes are joined together and you drive -ve cathodes separately - so you get terminals Common+, R-, G-, B-.

• The other controller expects common Cathodes with -ve Cathode end of LEDs common and +ves brought out so you get terminals common-, R+, G+, B+.

IF this is the case then the two systems are incompatible without either extra equipment.
In some cases cutting and rejoining of the internals of the LED strips may be posisble if they have used separate R, G, + B LEDs but in most cases the RGB is liable to be in one LED package and cannot be re-arranged.

If you can provide specific controller and LED strip brand and model references and ideally web links we should be able to suggest a controller design that would do what you want OR be able to suggest that you quit while you are behind.

Obvious in hindsight advice: When buying such equipment one should either know what is involved and be sure re compatability OR have a supplier guarantee that the components will interoperate and that you have right of return if this is not the case. (This is a lesson that your client needs to learn and you may or may not be in a position to "safely" teach it to them :-).)

From an electronic device schematic point of view, a fan simplifies thus:

Fan ==> Motor ==> Inductor coil(s)

Of course there are fans that do not use coils at all (piezo fans, coil-less motors), but let us set that aside for a moment.

At the lowest level of abstraction, thus, a purely electrical schematic symbol using one or more inductors could suffice to indicate a motor, such as these symbols for DC motors:

(Source)

Another source suggests the stepper motor symbols below: ()

At a higher level of abstraction, the familiar symbol(s) with the letter "M" in a circular enclosure are commonplace, such as the brushless and brushed motor symbols illustrated in the question.

At a more pedantic level, some schematics use the M-circle symbol with a variety of enhancements to specify the various types of motors. This page provides an entire flood of symbols for specific motors.

Note the use of both inductor and M-circle to address both electrical and functional symbolization in the 3-phase series motor above.

Even so, our symbols so far do not indicate the use of the motor for a fan. For fans specifically, there isn't any universally followed symbol AFAIK, but a few variants show up in various documents:

(clearly, a fan!)

(AC motor -> fan)

The "Other standards" fan symbols shown in the question are not so common in electronic design related schematics, they are used typically for process engineering and electrical wiring.

In summary, for lack of any universal standard, the symbol to be used is best determined by the purpose of the schematic: Is it important to account for inductive behavior, or to illustrate the type of motor, or at the most mundane and abstracted level, is it crucial to convey clearly that "This motor here is a fan"?