Both of those diagrams are connected to the grid, hence grid tied.
I'm not convinced, the grid tie aspect is synchronising the inverter output to the voltage and frequency of the grid so it can then increase voltage to feed appliances. Heres a wiki definition:
"Grid-tie inverters convert DC electrical power into AC power suitable for injecting into the electric utility company grid. The grid tie inverter (GTI) must match the phase of the grid and maintain the output voltage slightly higher than the grid voltage at any instant. A high-quality modern grid-tie inverter has a fixed unity power factor, which means its output voltage and current are perfectly lined up, and its phase angle is within 1 degree of the AC power grid. The inverter has an on-board computer that senses the current AC grid waveform, and outputs a voltage to correspond with the grid. However, supplying reactive power to the grid might be necessary to keep the voltage in the local grid inside allowed limitations. Otherwise, in a grid segment with considerable power from renewable sources, voltage levels might rise too much at times of high production, e.g. around noon with solar panels."
Its irrelevant where the solar power goes, wether to batteries or inverter, the invertor is tied to the grid.
As above
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Off grid is where there is no connection whatsoever to the property.
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I agree when you 'Off Grid' in terms of property, but if say it in terms of inverters you will have an AC IN, AC OUT etc, Off Grid for inverters means the output is isolated from the grid and output runs independently of the grid
This may be the reason the original OP had a problem with the company that gave the quote and what he was wanting, they could not do it as it was not an off grid system.
This is where I think the OP has got it wrong and a grid tied inverter would be much better