Why does ByteArrayOutputStream use int?

ByteArrayOutputStream is just overriding the abstract method declared in OutputStream. So the real question is why OutputStream.write(int) is declared that way, when its stated goal is to write a single byte to the stream. The implementation of the stream is irrelevant here.

Your intuition is correct - it's a broken bit of design, in my view. And yes, it will lose data, as is explicitly called out in the docs:

The byte to be written is the eight low-order bits of the argument b. The 24 high-order bits of b are ignored.

It would have been much more sensible (in my view) for this to be write(byte). The only downside is that you couldn't then call it with literal values without casting:

// Write a single byte 0. Works with current code, wouldn't work if the parameter
// were byte.
stream.write(0);

That looks okay, but isn't - because the type of the literal 0 is int, which isn't implicitly convertible to byte. You'd have to use:

// Ugly, but would have been okay with write(byte).
stream.write((byte) 0);

For me that's not a good enough reason to design the API the way it is, but that's what we've got - and have had since Java 1.0. It can't be fixed now without it being a breaking change all over the place, unfortunately.