If every application and system were properly color-managed it would be possible to get consistent color everywhere.
As it is, that's simply not the case.
Many folks feel that publishing an image in the sRGB IEC61966-2.1 profile is the best approach. There is debate about whether to publish a profile in the image to identify to systems/browsers that have proper color-management that it is using that profile - my opinion is that you should do so.
If you have a wide-gamut monitor that doesn't closely emulate the sRGB profile, and assuming you have it calibrated and profiled, you will see different things when you display an sRGB image using a fully color-managed application (e.g., Photoshop), a partially color-managed application (e.g., Internet Explorer), or a non-color-managed applications (e.g., an older viewer application). That is to be expected.
The various browsers implement different levels of color-management. I mentioned that IE only does half the job - it interprets image color profiles, but it assumes every monitor is sRGB regardless of its profile. Firefox can do a full job of color-management, even including non-image web page element color-management, but it takes reconfiguration to do so. Safari only does image color-management, not web page elements. Last I looked Chrome ignored color-management entirely.
Create your image in the sRGB color-space, trust what you see on your calibrated and profiled system when the image is displayed in Photoshop - i.e., make the colors look right there - and make your own choice about whether to publish with the sRGB profile embedded. That's about the best you can do.
Maybe for the web, but not for print.
That's what I was afraid of... an interpretation problem. How do you publish a profile within the image to identify to systems/browsers?
When you save a file there's a checkbox in the dialog in the Color section that you can check to instruct Photoshop to save a color profile in the image file (e.g., JPEG).