I'll speak about their grammatical differences, leaving their proof- and model-theoretic differences for someone more qualified to discuss. Each of these logics has a vocabulary $V$, which is the set of symbols out of which its well-formed formulas (e.g. terms, sentences) are generated. One usually singles out a subset of $V$ as the set of logical vocabulary $V_L$. It is these $V_L$s that distinguish logics at the ground level, making it very transparent which is an extension of which. Let's see:
$V_L$(PL) = { '$\lnot$' , '$\land$' }
$V_L$(FOL) = $V_L$(PL) $\cup$ { '=' , ' $\forall_1$ ' } where $\forall_1$ quantifies over individuals
$V_L$(SOL) = $V_L$(FOL) $\cup$ { ' $\forall_2$' } where $\forall_2$ quantifies over properties (of individuals)
$V_L$(HOL) = $V_L$(FOL) $\cup$ { ' $\forall_n$' } where $\forall_n$ quantifies over yet higher-order properties
$V_L$(TT) = $V_L$(_OL) $\cup$ { ' $\lambda$' } where _OL is a _-order logic (usually _ > 0)
Of course, each of these systems could be defined in different ways, choosing different sets of logical vocabulary. This is just one way of going about it. Now, as you already said, each of these logics extends the ones coming before it. With this vocabulary talk we can give precise meaning to that:
Def. Logic A is an extension of logic B iff $V_L$(B) $\subset$ $V_L$(A).
In the event that the converse doesn't hold, A is said to be a proper extension of B.
Lastly, for specific examples of differences, consider these formulas:
PL: '$\phi \lor \lnot \phi$'
FOL: '$\forall x (x = x)$'
SOL: '$(a = b) \equiv \forall P (P(a) \leftrightarrow P(b))$'
TT: $\forall x ([\lambda x. x](x) = x)$
Each of these sentences is also valid for logics following it (the other direction doesn't hold, of course). Notice that higher-order logic is left out, because there is no sentence $\phi$ s.t. HOL $\models \phi$ but SOL $\not\models \phi$, due to the fact that the power-set operation is SOL-expressible (Hintikka 1995).
For corrections/suggestions, please leave a comment or simply edit this post.
The internal logic of a topos is an instance of the internal logic of a category (since toposes are special kinds of categories). The internal logic of toposes (instead of an arbitrary category) can also be interpreted with the Kripke-Joyal semantics. For more on this, check part D of Johnstone's Elephant and chapter VI of Mac Lane's and Moerdijk's Sheaves in Geometry and Logic, lecture notes by Thomas Streicher, and of course the nLab articles on these matters.
I don't know the term "internal logic of a type theory". But check the (very accessible) introduction of the HoTT book on how type theory and logic are related.
The terms "internal logic" and "internal language" are often used synonymously. Personally, I prefer "internal language", since this stresses that one can use it not only to reason internally, but also to construct objects and morphisms internally.
The internal language of a topos $\mathcal{E}$ is higher-order in the sense that, because of the existence of a subobject classifier, every object $X \in \mathcal{E}$ has an associated power object $\mathcal{P}(X) \in \mathcal{E}$ which one can quantify over in the internal language. (In the special case $\mathcal{E} = \mathrm{Set}$, the internal language is really the same as the usual mathematical language and $\mathcal{P}(X)$ is simply the power set of $X$.) In an arbitrary category, power objects need not exist, such that their internal language is (at best) first-order. Generally, richer categorical properties allow you to interpret greater fragments of first-order logic, this is neatly explained in Johnstone's part D.
Any Lawvere-Tierney topology in a topos gives rise to a modal operator in its associated internal language. (These operators can have concrete geometric meanings, for instance "on a dense open set it holds that" or "on an open neighbourhood of a point $x$ it holds that".)
I don't know of a direct relationship between fuzzy logic and the internal language of toposes, see an older question here.
Best Answer
There are really two different meanings of "higher-order":
Referring to syntax
Systems that include variables for one or more types of "individuals" and also variabels for "sets" of individuals, "relations" on the set of individuals, or "functions" from individuals to individuals are called "higher order".
One common system for higher-order logic (with one type of individuals) is the following inductive definition of a large collection of "types":
The logic contains, for each type, a universal quantifier over objects of that type and an existential quantifier over objects of that type. As you can see, this looks just like a kind of type theory. One difference is that higher-order logic is more like classical logic than intuitionistic type theory. In particular there are no dependent types and no "judgments", just formulas and proofs that are analogous to the ones from first-order logic.
Indeed, all the syntax of this "higher-order" logic can be interpreted as just a multi-sorted first-order logic.
Referring to semantics
"Higher-order semantics", also called "full semantics" and "standard semantics", are a particular semantics for higher-order logic in which the "higher order sorts" are taken to include all the elements they possible can. For example, in full semantics, once the set of individuals $i$ is determined for a model, the collection of objects of type $P(i)$ in that model must be contain all subsets of $i$, and the collection of objects of type $i\to i$ must contain all functions from $i$ to $i$, etc. In this way, all the higher types in a model are determined solely by the set of individuals in the model.
In "Henkin semantics" for higher-order logic, the collection of objects of type $P(i)$ in the model might be a proper subset of the powerset of $i$ in that model, and the collection of objects of type $i \to i$ might be just a proper subset of all the function from $i$ to $i$, etc.
These semantics have very different properties. Full semantics, by more or less arbitrarily reducing the number of models, allows for categorical theories for the natural numbers and the real numbers. Henkin semantics have the same completeness and compactness properties as first-order logic.