The term "stream cipher" is much more general than how you understand it.

There is a huge number of stream ciphers that have never been broken.

There is also a great number of stream ciphers that have been broken, but almost all of them belong to a class of ciphers that deserves the name "cheap stream ciphers".

Such "cheap stream ciphers" have been designed so that they can have a much cheaper implementation than the ciphers that are based on pseudorandom functions similar in complexity to DES and its successors, with the hope that despite being cheaper they will still be hard enough to break.

In most cases, sooner or later these hopes have been shown to be unfounded.

The term "stream cipher" just means that it encrypts a stream of plaintext symbols by combining a stream of encryption mask symbols with them.

There are many kinds of stream ciphers based on how the stream of encryption mask symbols are generated and based on the methods used to combine them with the plaintext.

Even if more general combination functions are possible when the only condition is to be able to decrypt the ciphertext, when an additional condition is imposed, that the statistical properties of the plaintext must not leak into the ciphertext, then the encryption mask symbols must be combined with the plaintext symbols using a quasigroup operation (a.k.a. a Latin square operation).

Most frequently, it is preferred that the quasigroup operation is a group operation, because this makes it simpler to implement. Moreover, usually it is preferred that the group operation is an additive group operation, to have an even cheaper implementation. Such stream ciphers are called additive stream ciphers.

When implemented in software, there are many additive group operations with equal complexity. However, when the encryption is implemented in hardware, addition modulo 2 is the cheapest so it is preferred.

The use of addition modulo 2 as a combination method has been invented by Gilbert S. Vernam in 1918. The term "binary additive stream cipher" should have been synonymous with "Vernam cipher". Unfortunately, in another example of mistaken historical attribution, the term "Vernam cipher" is used to mean any stream cipher where the encryption mask stream is truly random, which is a different invention, first described by Frank Miller in 1882 and rediscovered and popularized by Joseph Mauborgne, a user of Vernam machines.

Based on how the encryption mask stream is generated, there are many kinds of stream ciphers, but most of the practical stream ciphers are either synchronous stream ciphers, where the initial state and the transition function and the output function of the automaton that generates the encryption mask symbols do not depend on the plaintext or on the ciphertext, and self-synchronizing stream ciphers, where the encryption mask symbols are generated either as a pseudorandom function of the ciphertext symbols (these are a.k.a. ciphertext autokey stream ciphers) or as a pseudorandom function of the plaintext symbols (these are a.k.a. plaintext autokey stream ciphers; they have been abandoned a long time ago, because they are less secure). The two kinds of self-synchronizing stream ciphers are dual, i.e. starting from one such cipher a dual cipher is obtained by using the encryption algorithm for decryption and the decryption algorithm for encryption.

The stream ciphers that are most frequently used nowadays are binary additive synchronous stream ciphers, but there is a great variety of possible structures for the automaton that generates the encryption mask symbols. When implemented in software on a modern CPU, the simplest such automaton is made by using AES in counter mode (CTR mode).