I think you've got the idea of it. In general, acceleration makes something be at a further distance from us as a function of redshift, hence the supernovae are dimmer (confusingly a magnitudes are defined bass-ackwards so that higher magnitudes indicate a dimmer object). You can see that in the curves. The red line has no acceleration, the black has some and the green has a lot more. Importantly the black curves shows deceleration at even higher redshifts, which you can just see in that plot, note how the black line is starting to flatten compared to the green by redshift 1. At higher redshifts this is more pronunced as the black line gets closer to the red again.
It is a good question, and one that a lot of work is done towards. In the local Universe we a pretty sure the SN are standard candles, but the high redshift Universe was a very different environment to today and since we don't really know much about the actual mechanism behind this type of SN it is possible that high redshift SN are just intrinsically dimmer, mimicking acceleration! This is somewhat unlikely because the spectra of these SN look the same, indicating that they do appear to be similar events.
The biggest non-standard candle-ness of SN can be nicely removed by a process known as 'stretch correction'. This where there is a relationship between the SN peak brightness and the light curve width (how long in days it shines for). Once the light curves are corrected for redshift (redshift stretches the light curve out in the same way as it stretches the light wavelength) they look like the top plot in the attached figure. Note that the peak brightness of them is all quite different. However, once a simple scaling is applied to the the brightness as a function of light curve width, the bottom plot is achieved. These stretch corrected light curves are then remarkably standard. Note that this stretch correction is an empirical result. It appears to work even though we don't have a great idea of the physics behind why it works. This is a gap in knowledge that a lot of theoretical work and low redshift SN observations are working to solve.