People are a bit loose with their definitions (meaning different people will use different definitions, depending on the context), but let me put what I would say. I will do so more in the context of modern computer vision.
First, more generally, define $X$ as the space of the input data, and $Y$ as the output label space (some subset of the integers or equivalently one-hot vectors). A dataset is then $D=\{ d=(x,y)\in X\times Y \}$, where $d\sim P_{X\times Y}$ is sampled from some joint distribution over the input and output space.
Now, let $\mathcal{H}$ be a set of functions such that an element $f \in \mathcal{H}$ is a map $f: X\rightarrow Y$. This is the space of functions we will consider for our problem. And finally, let $g_\theta \in \mathcal{H}$ be some specific function with parameters $\theta\in\mathbb{R}^n$, such that we denote $\widehat{y} = g_\theta(x|\theta)$.
Finally, lets assume that any $f\in\mathcal{H}$ consists of a sequence of mappings $f=f_\ell\circ f_{\ell-1}\circ\ldots\circ f_2\circ f_1$, where $f_i: F_{i}\rightarrow F_{i+1}$ and $F_1 = X, \, F_{\ell+1}=Y$.
Ok, now for the definitions:
Hypothesis space (HS): the HS is the abstract function space you consider in solving your problem. Here it is denoted $\mathcal{H}$. I find that this term does not appear very often in applied ML, rather, it is mostly used in theoretical contexts (e.g., PAC theory).
Sample space (SS): the sample space is simply the input (or instance) space $X$. This is the same as in probability theory, regarding each training input as a random sample instance1.
Parameter space (PS): for a fixed classifier $g_\theta$, the PS is simply the space of possible values of $\theta$. It defines the space covered by the single architecture that you train2. Usually it does not include hyper-parameters when people say it.
Feature space (FS): for many models, there are multiple feature spaces. I've denoted them here as $F_2,\ldots, F_\ell$. They are essentially the intermediate outputs due to the model's layered processing (but see note1). For CNNs, these "feature maps" at different layers are often used for different things, hence distinction is important.
For your example:
The HS is almost the same as the PS once you've chosen logistic regression (except that the HS includes the models arising from different hyper-parameters as well, whereas the PS is fixed for a given set of hyper-parameters). Indeed, here, the HS is the set of all hyperplanes (and the PS could be as well, depending on the presence of e.g. regularization parameters).
The sample space is the set of all possible cat images; i.e., $X$. It is not usually restricted in meaning to be $D$, which is usually just called the training set.
The feature space in your case is indeed $F_1 = X$, assuming that you feed the raw pixels to the logistic regression (so $\ell = 1$).3
1Some people treat some processed form of the input as the input. E.g., replacing an image $I$ with its HOG or wavelet features $u(I)$. Then they define the sample space $X_u = \{ u(I_k) \;\forall\; k \}$, i.e., as the features rather than the images. However, I would argue that you should leave $I\in X$ and simply set $F_1 = X_u$, i.e., treat it as the first feature space.
2Note that each $\theta$ defines a different trained model, which is in the HS. However, not all members of $\mathcal{H}$ can be reached by varying the parameter vector. For instance, you might search over the number of layers in a CNN, but the parameter space of a single CNN will not cover that. (Though note again that $\mathcal{H}$ tends to be used more in theoretical contexts). One distinction between HS and PS appears in the context of error decompositions of approximation vs estimation noise.
3Normally (in "older" computer vision) you would extract features from the image and feed that to e.g. logistic regression. The modern version of this is attaching a fully connected (linear) layer with a softmax at the end of a CNN.
Best Answer
For computer vision problems ConvNet produces remarkable success at recent years. It is a type of neural network and it doesn't unroll images into vectors as you mention. Hence, it can capture spatial relations between pixels better than fully connected neural network. Also it reduces number of parameters by using the same kernels (group of weights) all over the input space. It also known as LeNet named for Yann LeCun. If you are interested check his convnet papers. Also Alex Krizhevsky's paper on ImageNet at 2009 and consecutive papers describe how Convnet learns and what it learns.