OP:
"...All I want to do is to be able to edit my RAW photos in Lightroom 5, then use edit in > using PScc to do some final retouches in my Prophoto color working space, and then to save the final image as an Srgb to put up onwhich ever website it needs to go up on, and to look exactly like the original I was editing in LR5 and PSccwith no color changes. ..."
That's a very reasonable request. In fact, because of the well known benefits of staying in ProPhoto for as long as possible, what you described is exactly how I work most of the time, so it certainly can be done. The only caveat I should mention is that because sRGB has a vastly smaller gamut than ProPhoto, there is no way the two versions will ever look exactly the same, even if the conversion between color spaces is done properly -- there will always be some differences. Whether or not they are significant, that's your call.
With that one caveat out of the way, the crux of the problem is revealed in these three statements of yours:
(a) "The photo labeled Export has been exported in Tiff format with Lightroom adjustments, which I then resized and converted to jpg to reduce size.
(b) The other photo is a copy of the same Exported tiff photo which I opened in PS CC and converted to sRGB using the "Save for Web" option, and putting a tick in the boxes for "Optimised" and "Embed profile" (ProphotoRGB from PS CC color workspace) and using Bicubic resampling.
(c) You can see that the sRGB photo is darker with more saturated colors."
First, because what people see on their screen depends on what software they are using to view your images, because there is a bit of ambiguity in the way you describe them, and because people may not pay attention to file names, to be absolutely unambiguous, I'm going to simply refer to the two images you posted as #1 and #2.
Correct me if I'm wrong, but from your statement (a),
it sounds like you took no special measures to change the color space of this version, #1. So, I checked, and, exactly as expected, it is still in ProPhoto because that was your working color space. Because of this, and because not all browsers and image viewing programs can handle ProPhoto correctly, #1 is likely to look very different on different systems. You obviously don't want that.
Again, correct me if I'm wrong, but from your next statement, (b),
it sounds like in the first part of this statement you think that #2 should be in sRGB, but from the end this statement, you think that that #2 should be in ProPhoto. Obviously, only one of these possibilities can be true. A quick check (say, using Bridge) will show you that your image #2 actually was converted to sRGB. Because of this, it *will* display consistently in almost any image viewing program including pre-2000 legacy programs, web browsers, etc. This will be the way most people see your image (excluding differences in monitors, etc. that are beyond your control).
Your next statement, (c), is, "
You can see that the sRGB photo is darker with more saturated colors."
Well, because of the ambiguity of your 2nd statement, (b); because the sRGB version (ie, #2) actually looks lighter (not darker) to me; and, because I can't tell what you are actually seeing on your system, I'm really not sure which of the two images you prefer.
SO:
If you can tell me whether or not you prefer #2 (as viewed on the forum), we can proceed from there.
Also, since you mentioned you have Adobe Bridge on your system, we can do some tests using exactly the same software. So, view the two files that you posted in Bridge, and tell me whether:
(a) they look the same in Bridge, but when viewed as a forum post, you prefer #1; or,
(b) they look the same in Bridge, but when viewed as a forum post, you prefer #2; or,
(b) they look different in Bridge (and tell me which you prefer when viewing them in Bridge).
I'll wait to hear from you before proceeding.
Cheers,
Tom
PS - While I'm at it, let me respond to your questions about possibly using the Proof Tool, using MonitorRGB (for something - what?), and synchronizing color space settings between different Adobe products. The answer is simple: Don't worry about, or activate any of these options / procedures. At this point, they will only muddy the waters.