Basically there is only one definition of the general term "subobject" of $X$. It is an isomorphism class of monomorphisms into $X$. (Some sources remove the isomorphism class part and just take monomorphisms as the definition. Ok that makes two definitions then.)
It just turns out that in many categories this condition is much too weak. For example, in the category of schemes, the monomorphisms are the radicial formally unramified morphisms, but the term subscheme refers to something much more restrictive and adapted to schemes appearing in practice. Same for submanifolds. To remedy this, one may fix a class $\mathcal{M}$ of monomorphisms and define a $\mathcal{M}$-subobject to be an isomorphism class of morphisms into $X$ that belong to $\mathcal{M}$. See Definition 7.77 in Joy of Cats. One should assume that $\mathcal{M}$ contains all isomorphisms and is closed under composition, but this isn't always the case.
There are all sorts of general types of monomorphisms (see Remark 7.76 in op.cit. for a summary: iso => section => regular => strict => swell => strong => extremal => mono), but in concrete categories we can choose whatever fits our purpose, even though sometimes it just so happens that one of the general types already fits (subposets = regular subobjects). For example, a subscheme is a $\mathcal{I}$-subobject in the category of schemes for the class $\mathcal{I}$ of locally closed immersions. To answer the question in the title: Of course, this example does not change the general meaning of "subobject" for this category.
Many definitions can now be made relative to $\mathcal{M}$, for example the notions of $\mathcal{M}$-wellpowered category (Definition 7.82 in op.cit.), $\mathcal{M}$-injective object and $\mathcal{M}$-essential morphisms (Definition 9.22 in op.cit.), $\mathcal{M}$-partial morphism (Definition 28.1 in op.cit.) and $\mathcal{M}$-topos (Definition 28.7 in op.cit.), where a $\mathrm{Mono}$-topos is the same as a topos in the usual sense.
Having said this, it should be clear that even for a fixed category there is not a correct choice of $\mathcal{M}$-subobjects to look at. As Todd already commented, it depends on what you want to do with them. Also in the example of the category of all $\mathcal{U}$-categories, in some situations we need subcategories (which indeed are just subobjects in the categorical sense), but in others we need full subcategories, wide subcategories, or replete subcategories.
The story also dualizes, for example we get the notion of a regular-quotient, which captures what we usually understand as quotients of algebraic structures, and of a regular-projective object, which by definition is able to lift morphisms along regular epimorphisms. For any algebraic theory, free algebraic structures are regular-projective, but not necessarily projective since there might be too many epimorphisms.