"Well-founded" and "inductive" mean roughly the same thing. I think the reason different terminology tends to get used for W-types is that their definition looks similar to notation for ordinals (imagining that the branching of the trees is like a limit of the children). They're distinct from the other inductive types given in Martin-löf type theory in that they are potentially infinitely wide. But they're still supposed to be (intuitively) finitely deep, which is what the "inductive" and "well-founded" refer to
Whether or not you can encode various other sorts of inductively specified types as W-types is a complicated question.
It's pretty easy to show that you can encode plain inductive types using W-types if you have various sorts of extensionality. The typical schemas for what constitutes an 'inductive definition' are like finite polynomials, while W-types are trees built from any (infinite) polynomial definable in type theory. The main discrepancy is how the 'products' act. The finite products in inductive schemas have a canonical form, while the functions used in W-types do not, and the most obvious way to fix that is with function extensionality.
However, you can also get by without extensionality. You only need some eta rules. The idea is to use W-types to build the right sort of trees, then define the subtype of 'canonical' trees by (W-type) induction. It happens that the recursion rule for this subtype has the judgmental behavior matching the schema for inductive definitions.
You can also encode indexed W-types using normal W-types and the identity type using a similar strategy. You use W-types to build a larger type of trees on the 'total spaces' involved, and then define a subtype of well-indexed trees by induction. This file shows how to do it.
You can (I believe) encode indexed inductive types/families using indexed W-types, including the encoded version of those, using a strategy similar to the Why Not W? paper. My Agda file above shows how to do this for a fairly simple indexed type that was mentioned in another question here.
You can encode mutual inductive and inductive-inductive types with indexed inductive types. For mutual inductive types, you just add a finite index type to turn N definitions into a single N-indexed definition. For induction-induction, you follow a similar strategy as for building indexed W-types: define mutual inductive types that contain too many values, then define the subtypes with proper indexing afterward.
You cannot encode all inductive-recursive definitions as (indexed) inductive definitions. I-R definitions were invented as a schema that would let you write down the definition of universes as a special case. However, the additional power comes from simultaneously being able to define a recursive type family. If instead you just simultaneously define a recursive function into an existing type, I believe they are encodable using a strategy like above. This might mean that if you have enough universes, you can use them with inductive definitions to encode everything you could write by just having a theory that admits inductive-recursive definitions (but with no pre-specified universes). I'm unsure about this, though.
Having a universe that classifies inductive-recursive definitions is even stronger, and isn't itself an instance of induction-recursion. It's actually inconsistent to be able to do induction on such a universe (while the I-R definable universes in 6 can have an associated induction principle).
Quotient/higher inductive types can't be encoded as any of the previous sorts of definition in general. Quotients where you can compute a canonical representative for each equivalence class can be defined, but not all quotients are like that.
I don't think strict positivity has anything to do with being inductive/well-founded (negativity does, though; see the comments). It's necessary to guarantee that 'all' inductive definitions are meaningful in various sorts of models. For instance, you can't have a classical set theoretic model of an inductive type $T \cong 2^{2^T}$ (which is positive, but not strictly positive), because $2$ classifies the propositions, and you can't have a type equivalent to its double power type. Constructively you might be able to admit some such types, and for instance, $λ \_ → 0$ and $λ\_ → 1$ give you a starting point for building up your finitely-deep values. However, these sorts of non-strictly positive types can conflict with other features than just classical mathematics, so you need to be very careful.
Various bits of the above are subject to caveats about the details of what counts as an "inductive definition." There's literature out there rigorously defining various schemata for what constitutes an (indexed) inductive(-recursive/inductive) definition (etc.). Agda (for instance) is not super rigorous, and runs a checker that lets you conveniently do things that could probably be encoded in those more rigorous schemata in a more inconvenient way. Or perhaps you couldn't, but it's still fine; or isn't.
