You're asking how to adjust the glued up plumbing to match up with the new trap you've purchased. I'd suggest that the much easier option to is to return the new trap you've purchased and buy a different one that fits the existing glued up plumbing.
However, if you're set on using the trap you have in hand...
You cannot undo the section of piping identified by the red dot because those fittings have been glued together. There are techniques for undoing the glue, but people seem to have varying levels of success with those techniques, most, from what I've seen here, fall into the "little to none" category. Instead, you'll have to cut the pipes off. This will require a small hacksaw or pipe cutter. In either case, be sure to make your cut nice and square to the pipe, or you may end up with leaks when you reassemble new pieces.
You would have to replace this plumbing all the way to where it screws into the grey Air Admittance Valve (AAV) (it does screw into that, doesn't it?), or, possibly, by cutting the short straight section that has the actual dot on it.
You'd then recreate this creative bit of plumbing, but make it a little shorter so that the inlet end is lower to match up with the outlet of your new trap.
You might be able to get away with cutting the vertical white pipe between the two grey pieces (AAV on top and "dump trap"?) on the bottom. Shortening that vertical white piece would allow you to move the whole assembly down to meet up with the new trap.
I'm not sure what, exactly, you mean when you say that the new trap has a "horizontal nozzle" since it seems to me the existing one has a horizontal exit, as do all the ones I've seen installed in my house or at the store. However, you have the pieces in hand, you'll have to do the fiddling to make it fit.