>If we assume 670 GW of solar generation capacity was installed in June

Was installed /as of/ June. (sorry I was double checking your workings and got confused)

Assuming the figures aren't typos.

1) What is counted as generation capacity? a 'finished' solar farm that is waiting for paper work / connection to the grid could be included in the figures.

2) What about solar that doesn't see the grid? the panels on my roof might be included as installed capacity, but not show in generation figures.

edit: 3) another commenter mentioned solar with pumped hydro. Is that classed as pumped hydro in the generation figures?

Thank you for the correction! I didn't spot the ambiguity.

All three of your questions are good ones, and I don't know the answers.