How important is it to embed a colour profile within a JPEG for viewing on phones, pc, online etc?
I ask as the only way to seemingly preserve the exact colours of an OOC JPEG is to not convert or embed a colour profile? I realise info is lost etc, but the differences in colour shift aren't at all noticeable. If you embed default PS sRGB under export and embed, the picture will be different. Presumably this is due to minor differences in colour information being dumped/altered? Even having PS set to default preserve colour profile of an image when opening, if you export and just tick embed on its own, the image will still be different despite no conversion (it's already sRGB ooc). In an example image I took, the difference is noticeable with a shift to being cooler with more green/blue. The image also gets a fraction more washed out but that's much harder to spot. It's not 'bad' and not drastic, but it's noticeable. This is the case opening in firefox, when uploaded online etc.
This means if you want to crop ooc JPEG just once and do nothing else, the image will always be different unless you don't embed the sRGB colour profile the image has by default. It's then left without a colour profile.
Cropping in camera seems to crop the original JPEG as opposed to cropping from it's RAW file. There's no way to crop the RAW file in camera. This results in a fine jpeg becoming a normal quality jpeg with no way around it. In another example image, the in camera crop system (XT5) seems to have in built colour profile shift settings to try and compensate for the resaving of it's own JPEG whilst retaining a sRGB colour profile. In a different example image, it appears to add a fraction more warmth/red/magenta to compensate for what is presumably the shift to looking slightly cooler just like what I had with the first example image mentioned above in PS. Issue is, it seems to go a fraction too far the other way becoming warmer instead of the same. This could be inaccurate and it's simply a limitation of resaving a jpeg even just once and putting an sRGB profile over the top of an sRGB profile as the original sRGB profile interpretation of the RAW data is different to the reinterpretation of the JPEG data (it shouldn't make virtually any difference but it seemingly does, even in camera when it's using presumably the exact same sRGB profile settings).
Any way to keep sRGB embeded colour profile without any colour shift?