Sadly I'd like to know an answer for your first question as well.
For your second question, you just need to see the difference between the description of a protocol and an actual instantiation of it (meaning, a cryptographic scheme). Diffie-Hellman is a cryptographic protocol, describing a way for two parties to exchange a common element in fixed ambient abelian finite group $G$, with cardinality $p$ (usually, a prime number). The security of DH will then be essentially defined as the smallest amount of operations in the group you can do to break the protocol. Brute forcing is the the most natural way to go: enumerate all possible solutions and you'll find the correct one. At worst, you will need to enumerate all $p$ elements. These guys are represented by binary strings of size $\log p$, so the computer will take $p = 2^{\log p}$ operations, an exponential number in the size of the input. If we cannot do better, then this protocol is "secure" and gives you $\log p$ bits of security. Take $p \approx 2^{128}$ and you have a 128-bits secure key-exhange... That is if you could not do better than brute-force.
Let's say you do not have any information about the said group $G$ apart from computing its group law and testing for equality (this is an informal way of saying that you consider your group as a generic group). Then the best way to recover the shared element if you're not one of the two parties is to solve a discrete logarithm problem in $G$: given $g$ a generator, and $h = g^x$, compute $x$. I hope not to be burnt alive by people noticing that DLP and ECDH are not exactly the same problem, but for the sake of this discussion, let's say they are.
It is known (a theorem by Shoup, or Nechaev if you can read russian) that you will need (at least) $\Omega(\sqrt p)$ group operations to compute $x$. In fact, there are well-known algorithms that run in $O(\sqrt{p})$ to do that: Big-step-Giant-steps, or $\rho$-Pollard style random walks. These algorithms are therefore optimal in the generic group "model", and you can see that they require exponential time in the size of the group elements. To achieve 128 bits of security, you would need a $256$ bits prime.
Now in real life, you will instantiate $G$ to some actual group. For example, you could chose $(\mathbb Z/p\mathbb Z, +)$. However, the DLP here is trivial to solve: it's essentially extended Euclid algorithm which runs in $O(\log^2 p)$: this is quadratic in the size of the group elements, therefore you will definitely not do crypto with this: your prime would need to be absurdly huge to guarantee 128 bits of security (and then all inputs would be of the size of this prime).
But you could also select the group of rational points of an elliptic curve defined over a prime field: this is commonly defined as ECDH. In this latter group, in the current state-of-the-art, we do not know a better algorithm that the ones I gave you in the generic model (apart from very special cases that are irrelevant nowadays since they can be avoided easily). In other words, the best known algorithms to solve ECDH run in exponential-time. So compared to, say, DH over $\mathbb Z/p\mathbb Z$, it "gives more security", but notice that this is relevant to the group, not the protocol itself.
There are other possible instantiations. Invertible of finite fields, elliptic curves over extension fields, more complicated algebraic groups. Elliptic curves give a very nice combination of efficiency and security.
security-definition
tag from the question as it did not appear to be relevant - if you really want some kind of formal definition of some formal term, please include the request explicitly in the question. $\endgroup$