How is the term "solar system" defined? Could confirmation of a new planet lead to a change in this definition?
The definition of the edge of the Solar System is not one of particular scientific importance, so it is not particularly well-defined. There are a number of plausible definitions, all reasonable.  I've listed one which occur to me below in order of increasing size:


The pancake-shaped volume which contains the orbits of the major planets.
The sphere big enough to contain the major planets.
The volume inside the solar bubble.
The thickish disk which contains the Kupier Belt.
The sphere which contains the Oort Cloud.
The Sun's Hill sphere.
The irregularly-shaped volume of space where the Sun is the closest star.


As you note, the discovery of a Planet X would change some of them, but not others.

How choose to define the edge of the Solar System depends a lot on what you intend to use it for.  I very much doubt that any one will win out, at least not until the definition of the Solar System makes an important difference somewhere: government, commerce, science.

So I think there's no clear answer to your question possible.