*Recursive* vs. *inductive* definition
Both the terms "recursive definition" and "inductive definition" are common, and the differences between the terms are usually so insignificant that either one will work (so it's not worth losing sleep over). Some people are very fastidious about whether they call each definition "inductive" or "recursive", which is respectable. But I cannot immediately think of an example where changing from one word to the other would change what is going on in a particular definition.
My best description is that "inductive definition" is more common when we are defining a set of objects "out of nothing", while "recursive definition" is more common when we are defining a function on an already-existing collection of objects.
A prototypical inductive definition is the following definition of the set of natural numbers:
0 is a natural number
If $n$ is a natural number, so is $n$ + 1
Only numbers obtained from rules 1 and 2 are natural numbers.
Here we don't already have a set of natural numbers, we are describing a certain way to characterize which numbers are "natural numbers".
On the other hand, the following definition of the factorial function is usually called a "recursive definition" $$ n! = \begin{cases} 1 & n = 0 \\ n \cdot (n-1)! & n > 0 \end{cases} $$
The inductive definition of $\mathbb{N}$, which we already have, is what justifies the use of a recursive definition for $n!$. Any inductive definition leads to an induction principle and a method of defining functions by recursion.
It does matter that, in defining $\mathbb{N}$, we have to start with 0 and "work up". It doesn't work to try define the set $\mathbb{N}$ by saying that $n$ is a natural number if $n-1$ is a natural number. But once we have an inductive definition of $\mathbb{N}$ we can leverage that for other purposes.
A few other things that can be defined inductively are the set of polynomials over a ring, the collection of rooted finite trees, and the collection of Borel sets. Inductive definitions are particularly important in computer science, mathematical logic, and related areas. One additional aspect of them is that they usually give a set of "terms" for the objects being described. For example, we know from the inductive definition of $\mathbb{N}$ that every natural number can be expressed as a finite string of the form $0 + 1 + 1 + \cdots 1$ (of some length).
Wikipedia actually seems to distinguish between these definitions. I've actually never heard "inductive definition" being used, but I'll try to get the idea:
In your example however, it basically feels like thinking about the same thing in two different ways:
Recursive:
In order to compute $a_n$, first compute $a_{n-1}$ and let then $a_n = 2a_{n-1}$. Terminate when you reach $a_0=1$.
Inductive:
Start with $a_0=1$. Now if you know $a_n$, you can compute $a_{n+1}$ by $a_{n+1}=2a_n$.
That's pretty much the same! However one can think of cases where the recursive definition is obvious but the inductive one feels strange, namely whenever choices occur: Consider the stopping times for the Collatz sequence
In order to compute $s_n$ for $n\neq 1$, compute $$ s_n = 1+s_{f(n)} \text{ where } f(n) = \begin{cases} \frac n 2&,n \text{ even} \\ 3n+1&, n \text{ odd} \end{cases} $$ Terminate with $s_1=1$.
Now try giving a nice inductive definition ;)
In other contexts, the inductive approach feels more natural
If A is a valid term b is a variable, then $\lambda b . T$ is a valid term.
However, I wouldn't really bother the distinction, as one will get the point anyway. In the case of the sequence, what we acually mean is
Let $a_n$ be the unique sequence that satisfies the recursive equation $$a_n = 2a_{n-1}, a_0=1 $$
Nobody would call this an inductive equation, right?
Recursive definitions are technically unrestricted, whereas inductive definitions must usually have a well founded "induction principle" which actually lets you do induction (in the proof sense) on the object. Recursive definitions don't a priori give you inductive definitions, but an inductive definition is recursive.