EtBr and SYBR green are intercalating agents, that will place between the base planes of your DNA, which introduces negative supercoils in it. There is a limit to how much EtBr can be intercalated on supercoiled, closed circular DNA, due to these negative supercoils, but not on open circular or linear DNA (this was formerly used to purify supercoiled DNA on density gradients). This influences migration on the gel, if you add the EtBr to the sample before loading, or into the entire gel. EtBr and CYBR Green probably affect this differently, as would different concentrations of EtBr. This can explain differences in relative mobility.

EtBr and SYBR green are intercalating agents, that will place between the base planes of your DNA, which introduces negative supercoils in it. There is a limit to how much EtBr can be intercalated on supercoiled, closed circular DNA, due to these negative supercoils, but not on open circular or linear DNA (this was formerly used to purify supercoiled DNA on density gradients). This influences migration on the gel, if you add the EtBr to the sample before loading, or into the entire gel. EtBr and CYBR Green probably affect this differently, as would different concentrations of EtBr. This can explain differences in relative mobility.

it follows from @Marc's answer that if you run the gel in the absence of intercalating dye then post stained the gel that the expected pattern would be produced ( and the gel would run a bit quicker (10%))

Is the issue primarily caused by co-loading the dye into the samples instead of loading it into the agarose? The dye was added prior to the run in both situations, though the first run had dye added directly to each sample.

yes as Marc says pre adding dye ( sybr or etbr) or having dye in the gel will both allow rapid intercalation of the dye and produce altered running speeds. Try just adding orange G or xylene cyanol as a viewing dye and glycerol aa the loading dye mix and running in a gel without etbr or sybr then staining by soaking the gel in dilute etbr or sybr to visualise the dna when the run has fininsed

Yes, as Paul stated previously, both adding the dye to the sample before loading on the gel, or having the dye in the gel would lead to intercalation during the run. The only way to get rid of that is to run the gel without dye, and stain it after the run.

One important point here coming back to the original question: adding the dye to the sample would probably be at a higher concentration than adding it within the gel, thus leading to different effects on mobility. Also, different dyes may have different effects.

Also, important to realise that the effects of the dye are different depending on the form that you are looking at: linear and open-circular DNA gets larger upon intercalation, closed circular will first get larger (at increasing dye concentrations) because the negative supercoils counteract the positive ones that are naturally present, but at higher concentrations, it gets smaller again. So the relative migration pattern may change with dye concentration and dye. Just stick with one protocol.