What's the difference between "People are all doing ..." and "People are together doing ..."

There is the question that the difference on these meanings below.

When we are in a dinner, then I have found a couple who eating a dish. I say that:

1. That couple are both eating a dish.
2. That couple are together eating a dish.

These could be the same meanings, but I have found the meanings of "together" on Oxford Learner's Dictionaries

1. with or near to somebody/something else; with each other
2. so that two or more things touch or are joined to or combined with each other

I suspect that the meaning of "together" is only physically close to the people who are in a context/sentence. What is "both" for?

Would you help me on this question?

Thanks, Yuichi

The fact that the examples used in the question involve a couple and eating confuses the matter. The difference between both and together is obvious if one considers a different pair of examples, such as:

Jack and Jill are both doing their homework.

Jack and Jill are doing their homework together.

The second logically entails the first, but not v.v. The first, but not the second, is compatible with their working separately in their respective rooms on the items of homework that have nothing to do with each other. The second would most likely be used in a situation in which the two of them are collaborating on the same homework; at a minimum, it implies that they are in the same room, and that they are in some way supporting each other in their work.

The question also asks 'What is "both" for?' in the sentences of the first type. Both is semantically redundant: 'Jack and Jill are both doing their homework' is true if and only if 'Jack and Jill are doing their homework' is true. Both nevertheless does have a purpose: it is an emphasising device.