MacOS – Can anyone explain to me the difference between terminal apps, CLI, CLT and Prompt

Adding to the excellent answer by Steve Chambers:

Historically, the term Terminal was used to refer to a communication device that lets user interact with a centralized computer. A terminal provided an input/output device in the form of a keyboard and display.

The days of mainframes with remote access are no more, and, with almost everyone having access to a fully capable computer of their own with dedicated CPU and memory, usage of the term Terminal is now a misnomer, and thus, it's more appropriate to use the term Terminal Emulator as the Terminal.app is emulating a terminal device and the same macOS operating system which is running the Terminal.app is also running the shell program with which the user actually interacts.

Thus, in the current age where we no longer use real terminals, the term terminal refers in short to a Terminal Emulator.

macOS comes with a built-in terminal emulator called Terminal.app, however, there are 3rd party alternatives available with extended feature set. One such very popular 3rd-party terminal emulator program is iTerm2.

Command Line Interface or CLI for short is a general term to refer to a computer user interface where the primary mode of interaction is via typing commands which adhere to a specific syntax. It is in contrast to the Graphical User Interface or GUI which is prevalent in modern day computing devices.

While macOS comes pre-installed with a variety of UNIX command line tools, Apple's Command Line Tools or CLT is the term used to refer to a specific installable package made available by Apple for software developers. CLT, in addition to the UNIX tools pre-installed in macOS, installs macOS SDK, headers, and build tools such as Apple LLVM compiler and Make. It enables developers to build and install open source software. For details, refer to this link: https://developer.apple.com/xcode/features/

It's worth noting that the term Command Line Tool can be interpreted in two ways. Command Line Tools is a software package made available by Apple which contains a specific set of items. Command Line Tool(s) in general could be used to refer to any tool/app which is interacted with via a command line interface (Homebrew is latter of the two). And any command line app can be in general be referred to as a command line tool. (Just like how Apple uses the term App Store to refer to their marketplace for iOS apps, while the same term could also be used in a generic way).

Prompt or Command Prompt is another commonly used term to refer to the command line interface. A prompt is generally used to indicate the location for users to input command line (i.e. sequence of characters describing the entire command). A command prompt may use a special character to give some idea about the shell in use or the logged in user. E.g., BASH shell uses $ for the prompt character, while ZSH uses % for the prompt character. Prompt for the root user is almost always #. A prompt is generally followed by a blinking cursor to indicate the place for text entry.

Here's a depiction of a command prompt running BASH shell in Terminal.app in macOS for a non-root user. The part underlined in green indicated the prompt in the shell which is accessed by the user using the Terminal app:

To answer your specific questions:

The thing that threw me off was realizing that Apple does not ship macOS with CLT (Command Line Tools) and I have to download it.

macOS doesn't ship with Apple's CLT because most users of Mac (even most software developers) would never ever need it. And once you have a need for Apple's Command Line Tools, Apple has made it very easy to install them.

Doesn't that mean that Terminal is useless unless I download CLT?

Terminal is very much useful and usable for a variety of tasks, even for many software developers even without Apple's Command Line Tools installed. Most software developers would never install Command Line Tools and yet they can access any one of the built-in shells, run shell internal commands, execute shell scripts, run one of the numerous, pre-installed external commands etc. Only for the developers needing access to specific tools provided by CLT, would they need to obtain them by installing it.

Is it a compiler? And if it is built in macOS, why do I need to download Xcode's CLT?

No CLT is not a compiler, or aptly speaking, it contains a compiler (Apple LLVM) as one of the apps in its package of tools. Command Line Tools have been aptly described by Apple here:

Why do I need an app like Terminal to access the CLI. Isn't the CLI an interface to begin with?

Installing Apple's Command Line Tools installs a package containing header files, SDKs and various command line apps in macOS. Terminal.app is then used as an interface to gain access to the tools themselves.

Does Homebrew (which is a package manager) considered a CLT that has its own CLI that I interact with using Terminal?

Homebrew is a community developed, open source, 3rd-party package manager which makes it simple to manage installation of command line apps/utilities which Apple doesn't ship. It takes away the hassle of compiling/installing/configuring/managing/uninstalling a variety of open source command line utilities, which could be a hassle for a majority of users (software developers included).

Homebrew itself becomes available to users as a external command itself that lets them manage other command line utilities. It is an app/utility which is different from Apple provided Command Line Tools.

(To further clear any confusion, when referring to Apple's Command Line Tools, it is implied that one is talking about the specific package containing multiple tools made available by Apple, though the term itself sound pretty generic (like other terms Apple use such as App Store). Apple's Command Line Tools should not be confused with command line tools when referenced to in general)