The Commission is formed by commissioners, each of which has been nominated by a different nation's government. The Council consists of actual members of the national governments.
The third party that can't propose legislation but has to approve it (and which strongly opposed this) is the Parliament, which is directly elected.
OK, this is probably some nuance of the English language that I'm missing as a non-native speaker, but I meant that the people that make up the Commission are not part of the national governments. The people that are part of national governments each get to nominate one commissioner though, in addition to being part of the Council.
The third party that can't propose legislation but has to approve it (and which strongly opposed this) is the Parliament, which is directly elected.