From which one sees that the DC is derived from incoming AC - via a suitable rectifying (DC generating) cct. - to charge up the battery. The battery only comes in to play/service when there is either loss of incoming AC; or a significant fall/rise in AC volts in?
From the above article one can see that although there is a design that has the battery in cct "all" the time - there are designs where the battery is "only" actively in cct. when the incoming AC fails in some way or other.
The battery o/p in the first model is "always" supplying DC to the inverter to provide the AC o/p. In the other models the battery provides DC to the inverter "only" when required.
The term converter could be applied (or perhaps misapplied) to an inverter - since an inverter does in effect "convert" DC to AC...?
Equally the term converter could be applied to the DC charger cct/module; since it too converts - in this case AC to DC...
The inverter requires DC to function - regardless of the design of the UPS. In the first model/design if there is no DC there is no AC o/p. And since the battery is always in cct . this can only occur if the battery either has failed or isn't charging.
In the other model(s) the battery is providing DC to the inverter "only" when incoming AC fails...
If the symptoms are no DC - and the UPS is the first model, and one does not apply the term "converter" to the charger cct./module, then the first place to check is the battery - as it's the interface between input AC and out AC; since it provides the required DC to the inverter/converter?
If the symptoms are no DC and the UPS is one of the other model(s), and again one does not apply the term converter to the charger cct/module, then again the first place to check is the battery - the DC source into the inverter/converter?
If however one applies the term converter to the charger cct/module then that is the first place to check - as it's the primary DC source for all models of UPS... Be they the design where the battery is always in cct. (first model); or only in active cct. when required in the other models.
It would seem that the ambiguity of the question is deliberate? Either that or poorly thought out and presented. A clear definition of terms is missing and that leads to the konphusions noted...
Overall... since the battery is the key/essential item when the UPS is called to act (when no AC incoming etc..); and the battery provides the DC required to continue an AC o/p for whatever time; presumably the battery is the first place to check?