You said:
If you have any kind of load panel between the output of the battery system and the "line side" connection to the utility, you do not have a line side connection.
In other words you described
Service conductors <---> Distribution Panel <---> Solar and/or ESS equipment
I agree that's not a line side connection for the Solar/ESS.
Also it's not possible to provide backup power to the Distribution Panel in this arrangment.
But the post you responded to, I believe, was describing the following:
Service conductors <---> Solar and/or ESS Equipment (with MID) <---> Distribution Panel
In this case the power source equipment is connected directly to the service, which one might say is a line side connection, and the distribution panel can be backed up. The 'Solar and/or ESS equipment with MID' might merely contain connections for inverters (e.g. the Enphase System Controller or Tesla Gateway) or it might have the inverter in it with DC coupled battery (like an Outback or Sunny Island or SonnenCore).
In either example there could be splices to another set of service entrance conductors upstream, not shown, to another normal service disconnect that isn't backed up. For simplicity's sake I've also left out the location of the service disconnect and overcurrent protection, but that doesn't affect the issue under discussion. It might either be a separate fused disconnect in between, or part of SUSE equipment (Distribution Panel or Solar/ESS equipment in each example respectively).