reference frames - A few questions on passive vs active Lorentz transformations

1.) How do we physically interpret an active Lorentz transformation? The passive transformation seems simple enough: you view a fixed object from the perspective of a new observer. When we actively Lorentz transform a vector are we interpreting this as moving the vector to a new point in spacetime considered from the perspective of a single observer?

2.) I am reading David Tong's QFT notes, and am having a hard time interpreting what he means by active transformations. The notes in question can be found here: http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf The notes in question are on pg 11-12 as labeled by the book or pages 17-18 as labeled by the PDF.

In his notes, Tong states that we can transform a scalar field as follows:

$$\phi (x) \rightarrow \phi'(x) = \phi \left(\Lambda^{-1} x \right).$$

When he does this, I'm interpreting this as

$$\phi (x) \rightarrow \phi'(x') = \phi \left(\Lambda^{-1} x' \right),$$

where $x'=\Lambda x$. From what I understand, the advantage of using the inverse Lorentz transformation on the primed system is that we can use the same functional form of $\phi$. However, when moving to the primed system we have still used $\Lambda$, not its inverse. Can anyone tell me if I'm correct in my understanding up to this point?

If my understanding is correct up to this point then I really don't understand the next section in his notes. He states that under this transformation derivatives transform as

$$(\partial_\mu \phi) \rightarrow \left( \Lambda^{-1} \right)^\nu_{\phantom{\nu} \mu} (\partial_\nu \phi)(y),$$

where $y=\Lambda^{-1}x$ (where this $x$ is primed, right?). But we've still gone from $x \rightarrow x'$ where $x'=\Lambda x$ (again based on my understanding which may be terribly wrong). Using $\Lambda^{-1}x'$ was simply a mathematical trick to allow us to use the same functional form of $\phi$. If that's the case, why are the derivatives transforming as $\Lambda^{-1}$ instead of just $\Lambda$?

I'm sorry -- I know this is a little convoluted, but I'm having a really hard time getting my head around this, especially with his notation. I really wish he would have used primes or something...

Am I completely lost? Someone please rescue me.