thereby placing a significant limit on the programs that can be written for CDF?

I don't think it's much of a restriction since one can use Needs[...] in the Initialization option to load any packages that one wishes to use.

Why is that a bad thing? Because it causes repetitions and/or overwriting?

No, mainly because it is a hack (I say this lovingly) and FullDefinition is a function that was introduced in (and I believe not changed since) version 1.0. There is a new undocumented function called Language`ExtendedFullDefinition in version 10 (and perhaps 9?) which is possibly being used now for SaveDefinitions, though I have no real evidence for this.

When one has relatively little code to include in a Manipulate (or a DynamicModule) it is sensible to place the code directly in an Initialization option. However, once one had more than a modest amount of such code, this approach becomes non-maintainable, so Using SaveDefinitions becomes pretty important. Notwithstanding the dangerous cases that the gurus at StackExchange point out, it is generally safe to use. One thing that they point out, and which is true, is that a saved definition in a Manipulate may overwrite another definition of a function when that Manipulate is displayed on the screen. This also can happen with an Initialization option I believe. One way around this is to create one's Manipulates in a "unique" Context. Some other dangerous issues that are pointed out involve situations that the general user may not easily come by, but it is certainly important to be aware of them. It's kind of like how a physicist uses mathematics. her or she knows about things like uniform convergence of series, but does not worry too much about it in their day-to-day calculations unless a symptom of needing to look into things from this perspective rears its head. Then they take out their quiver of mathematical wisdom and address the issue. That's sort of my way of doing things.

One thing that is worth noting is that any functions or Symbols that one wants to be imported into the Manipulate or DynamicModule using SaveDefinitions must not be Protected (I just was reminded of this by looking though my email archive): this is presumably to keep many packages and contexts that are shipped with Mathematica from being placed in the Initialization option via SaveDefinitions.

It's true though (at least before version 10--I am not sure about any changes in version 10 since I haven't asked recently) that behind the scenes SaveDefinitions is makes use of FullDefinition to get it's information. This may have been upgraded in the recent version of Mathematica....

It took a little while ;-) but there were some clear symptoms. And, one way to have seen the issue (though I figured it out before this, but I was going to try this next) is that, if you call the manipulate output myManipulate and then execute

InputForm[myManipulate]

you get the following result (in the original problematic case):

Manipulate[If[\[Theta]1 < \[Theta]2, GraphicsRow[{gz, gw[\[Theta]1, \[Theta]2]}, Background -> bgColor, 
   Spacings -> 0, ImageSize -> Large], warning], 
 Style["Choose 1st and last points of domain.\nTry [0, \[Pi]/2] and [\[Pi]/2, 2\[Pi]].", Bold, 
  16], Delimiter, {{\[Theta]1, 0, Style["First", Bold, 12]}, 
  {0, Pi/12, Pi/6, Pi/4, Pi/3, (5*Pi)/12, Pi/2, (7*Pi)/12, (2*Pi)/3, (3*Pi)/4, (5*Pi)/6, 
   (11*Pi)/12, Pi, (13*Pi)/12, (7*Pi)/6, (5*Pi)/4, (4*Pi)/3, (17*Pi)/12, (3*Pi)/2, 
   (19*Pi)/12, (5*Pi)/3, (7*Pi)/4, (11*Pi)/6, (23*Pi)/12, 2*Pi}}, 
 {{\[Theta]2, 2*Pi, Style["Last", Bold, 12]}, {Pi/12, Pi/6, Pi/4, Pi/3, (5*Pi)/12, Pi/2, 
   (7*Pi)/12, (2*Pi)/3, (3*Pi)/4, (5*Pi)/6, (11*Pi)/12, Pi, (13*Pi)/12, (7*Pi)/6, 
   (5*Pi)/4, (4*Pi)/3, (17*Pi)/12, (3*Pi)/2, (19*Pi)/12, (5*Pi)/3, (7*Pi)/4, (11*Pi)/6, 
   (23*Pi)/12, 2*Pi}}, ControlPlacement -> Top, 
 FrameLabel -> Style["Multifunction Domain Selection", 20, Bold], Initialization -> Null]

And there is the smoking gun in the form of Initialization -> Null.

By the way, another way to handle getting this much code into the Manipulate without having to copy it all into the Initialization option, is to have the code external to the Manipulate and then five the Manipulate the SaveDefinitions->True option. This will automatically place the needed definitions into an internal Initialization option.

Finally, you probably want to update the notebook that's on your https://faculty.unlv.edu/gbp/mma/VisualComplexAnalysis/CDF/multifunctionCDF.nb page to the updated Initialization option as a delayed rule.

I''m glad this all worked out! --David

J Robinson, CDF is not a server side technology (if you know this already, my apologies). The CDF is downloaded from the server and plays in browser using the CDFPlayer plugin--executing on the local CPU.

What OS is on the remote server? Did you check if anti-virus is preventing execute permissions?