Help Critique deck post and beam location

The rule, at least here in the US, is a cantilever can extend no more than 1/3 the supported length (so, you could go ~3 feet further). Personally, I try to keep cantilevers no more than 2 feet, regardless of total supported length: while it might be OK structurally, it will likely sag the further you go, and can feel bouncy.

To do the best job, get some 12 foot 2x8s and sister them with the existing joists (run them the entire length, nailing with galvanized nails at the top and bottom every 16 inches). Then put a double joist hanger at the ledger against the house.

One of the critical things with a deck cantilever is how you attach the railing posts at the end of the cantilever. You will be nailing a 2x8 rim joist at the end to finish the deck, but you need to make sure the posts for your railing are tied in securely to the joists, and not just the rim joist. The method can vary depending on whether your existing railing posts are notched into the decking or fastened to the outside of the deck.

To accomplish the join I'd acquire some 1/4" by 3-1/2" by 24" steel plates and let them into opposing sides of the posts to act as gussets. I'd use countersunk flat-head sleeve bolts, piloted accurately and minimally, to sandwich the whole works.

Be sure to consider whether the existing posts are adequately anchored, as there will be substantial torque applied to them at their bases. It doesn't matter how well you attach the extensions to floppy posts.

That said, I think your plan seems flawed. A pergola creates some fairly unique structural concerns, namely the lack of diagonal bracing that a conventional roof would provide, as well as increased wind drag.

I'd be replacing your 4x4 posts with 6x6 posts that run all the way through. They'll do the job much better than spliced 4x4s, and they'll probably be more appropriate from an aesthetic standpoint, depending on your design. You don't want chicken legs under your nice new pergola.

If you elect not to do that, consider substantial diagonal bracing below the ceiling line connecting the posts to the primary horizontal members of your design in appropriate directions (where connection to a fixed structure doesn't provide support).