I'm currently studying splitting field extensions in Galois theory, and the book I'm using for a reference keeps saying that there are always field extensions $\mathbb{F}'$ of a non-algebraically closed field $\mathbb{F}$ which 'contain' $\mathbb{F}$.
Is this a sloppy use of language where the author is actually talking about a canonical field homomorphism $\phi:\mathbb{F}\rightarrow\mathbb{F}'$ with $\ker{\phi^{-1}}=\mathbb{F}'\setminus\phi(\mathbb{F})$ which serves as an embedding, where $\ker{\phi^{-1}}=\{y\in\mathbb{F}':\nexists x\in\mathbb{F}\big[\phi(x)=y\big]\}$, or is $\mathbb{F}$ legitimately included in $\mathbb{F}'$?
This is confusing to me primarily because the machinery through which the author constructs $\mathbb{F}'$ over $\mathbb{F}$ is as a quotient ring by a maximal ideal $\mathcal{I}$ in $\mathbb{F}[X]$, so $\mathbb{F}'=\mathbb{F}[X]\setminus\mathcal{I}.$ In this case $\mathbb{F}'$ definitely does not "include" $\mathbb{F}$ in any non-lazy sense, since the objects in $\mathbb{F}'$ are second order objects over $\mathbb{F}[X]$.
Despite this, there is some later discussion about the (non)uniqueness of algebraic closures in the sense that there are many isomorphic but non-equivalent sets that form the algebraic closure of a given field, primarily as a result of the non-constructive process of algebriac closure. This makes me wonder -- is the author implicitly assuming that $\mathbb{F}'$ is actually an isomorphic copy of $\mathbb{F}[X]\setminus\mathcal{I}$ which has the elements of $\mathbb{F}$ as first order objects contained in it?
I believe that this is a concern somewhat outside the usual domain of things field theorists care about, so I have added the model theory tag as it seems more appropriate to the question at hand.
This question is hardly restricted to the context of field theory: it is found throughout all of mathematics. Basically, whenever you have a chosen embedding of one set into another, it is extremely common to pretend it is actually a subset. For instance, using any of the standard constructions of the real numbers from the rational numbers, $\mathbb{Q}$ is not a subset of $\mathbb{R}$; instead there is just a canonical embedding $\mathbb{Q}\to\mathbb{R}$. Yet in almost all contexts, we say $\mathbb{Q}\subset\mathbb{R}$, identifying it with its image under the embedding. You can think of this as a sloppy use of language, as you say, but it is a sloppy use that is completely standard and used constantly in math.
In the particular case of your author, I don't know whether they intend to be talking about embeddings, or intend to be modifying the sets so they are actually subsets. For most purposes, if you want to be strictly rigorous, it is more convenient to define everything in terms of embeddings, so when you say something "contains $\mathbb{F}$" you really mean there is a chosen embedding of $\mathbb{F}$ into it. But I have no doubt that for everything they do, it doesn't actually matter.
As for the point about algebraic closures, I don't know what discussion you're talking about or what you mean by "isomorphic but non-equivalent". I'm guessing that the point of that discussion is that a field can have algebraic closures which are isomorphic as field extensions, but there is no canonical choice of isomorphism between them. Moreover, two different algebraic closures may have additional natural structures (besides just being field extensions of your base field) which are not isomorphic to each other. None of this has anything to do with the technicalities of underlying sets you are worrying about.