Does this mean that all agree or that the majority agree thus implying in the latter that some disagree?

I'd say the latter though I wouldn't agree with the implication. The others might have no opinion. Hope the following helps:

http://onelook.com/?w=consensus&ls=a

It could also imply a 'mutual feeling of general agreement' as in 'the consensus seems to be favourable'. But yes, a majority nonetheless.

Some places call it a 'group conscience': all in accord to some degree.

In short form voting, when concensus is determinative, the moderator will usually say something like this: "Are there any questions? any objections?" If there are none, he proceeds to say, "So ordered." A majority is presumed.

I'd go with all agreeing, at least to some extent IRL there would undoubtedly be some compromising required to come up with a final decision on which there is consensus and there may be some who give their individual consent grudgingly but still I would say there must be no members of the group who are adamantly opposed to the final decision..

I vote with Faldage.

Some dictionaries give different nuances in their definitions:

Dict.com:con⋅sen⋅sus 
–noun, plural -sus⋅es.
1. majority of opinion: The consensus of the group was that they should meet twice a month.
2. general agreement or concord; harmony.

Nr. 2 obviously offers a better foundation for long term agreement.

I would substitute "orally" for "adamantly." If there is any expressed opposition, there is no concensus. If there is expressed opposition, a vote of some sort must be taken. Then, unless the vote is a counted and recorded vote, the minutes could/would simply reflect that the motion had carried.

Solid overview and excellent nuance by PastorVon with "orally". After all, one usually hears consensus just as a decision is being made or has been made.

"So do we have a consensus about the word 'consensus' yet?"

I'm on board.