It seems to be "known" but not written down that the following are more-or-less equivalent:

pivotal (and spherical) categories
spiders, as defined by Greg Kuperberg
planar algebras, as defined by Vaughan Jones
non-symmetric modular operads, following Getzler & Kapranov

Then there is also the notion of a $T$-operad where $T$ is a cartesian monad. This was originally defined by Burroni and is the central concept in "Higher operads, higher categories" by MO contributor Tom Leinster.

My question is wether the first example can also be understood using the general theory of $T$-operads?

I don't have any good reason for asking. It is more that my head starts to hurt after contemplating this for any length of time.

For a more specific question: there is a forgetful functor from pivotal categories to directed graphs. I believe this is monadic and that the associated monad is cartesian. If so, we can consider the associated theory of $T$-operads and $T$-multicategories. However, so far, I have not been able to penetrate this definition.