Yeah, pretty much that.
Terminology upsets pedants like me because they actually confuse the layman of what’s actually going on.
The “industry” has sold us products using the terms “Inverter”, “Solar Inverter, “Hybrid Inverter”, and “Battery-ready Inverter”. In doing so, they’ve created a “Confusopoly”.
In a pure sense, an “inverter” is simply a device that converts DC to AC. (The reverse is a “rectifier” that converts AC to DC)
In a household solar power perspective, what’s sold as an “inverter” is a device that incorporates a solar MPPT tracking controller which feeds in to an inverter that takes the say 80-600V DC from the panels, inverts it to 230V AC, syncs the frequency, and applies electronically any other controls such as max export.
A device that’s sold as a “hybrid inverter” is similar but it’s designed to work with a battery. The difference is that the “inverter” in these devices is actually an “inverter/charger”. The inverter has additional circuitry and smarts to actually work in reverse - to charge a battery. In order to charge the battery, it works in reverse and then rectifies the AC to DC and charges the battery.
When you or it wants to discharge the battery to provide AC power, it’s just an inverter again.
Translating that to popular products:
A Fronius 5.0-1 is a device with a solar MPPT controller and an inverter. You can plug solar panels into it, but not a battery.
A Tesla PW2 Is a device with an inverter/charger that can connect to a battery (only the in-built battery). It does not have a solar MPPT charger built into it, so you cannot connect solar directly to it.
A Fronius Gen24 is a device that does everything a Fronius 5.0-1 could do plus it incorporates an inverter/charger instead of just an inverter. So it can connect directly to a battery. (But it will only work with compatible batteries eg BYD)
A Tesla PW3 is similar but it incorporates its own battery. It differs from PW2 in that in includes 6x MPPT solar inputs, thus removing the need for an extra device.
EDIT: an “inverter” doesn’t just change DC to AC, it changes the voltage in the process, to what is required. The “inverter/charger” also adjusts the voltage when it’s operating in reverse. To what is required to charge the battery. It’s all microprocessor controlled to what’s required.
Disclaimer: this is my understanding - please feel free to correct anything, I am not an expert