Government means precisely just the executive branch in international English across most polities with institutions.

That's not what the constitution says though. "Government" ("Gobierno") is what an American would understand "executive branch" to be, I'm guessing this is why it's confusing. I tried to make it easier by adding the translations, but maybe that's just making it more confusing :)

I guess broadly in English you'd say the judges are part of the state, but they're not a part of the Spanish Government.

That’s true in America, but the word government is applied more narrowly elsewhere, including in the UK.

The judiciary is part of the state. The government is also part of the state. They are different parts.