If you submit the form via JavaScript (i.e., formElement.submit() or anything equivalent), then none of the submit buttons are considered successful and none of their values are included in the submitted data. (Note that if you submit the form by using submitElement.click() then the submit that you had a reference to is considered active; this doesn’t really fall under the remit of your question since here the submit button is unambiguous but I thought I’d include it for people who read the first part and wonder how to make a submit button successful via JavaScript form submission. Of course, the form’s onsubmit handlers will still fire this way whereas they wouldn’t via form.submit() so that’s another kettle of fish…)
If the form is submitted by hitting Enter while in a non-textarea field, then it’s actually down to the user agent to decide what it wants here. The specifications don’t say anything about submitting a form using the Enter key while in a text entry field (if you tab to a button and activate it using space or whatever, then there’s no problem as that specific submit button is unambiguously used). All it says is that a form must be submitted when a submit button is activated. It’s not even a requirement that hitting Enter in e.g. a text input will submit the form.
I believe that Internet Explorer chooses the submit button that appears first in the source; I have a feeling that Firefox and Opera choose the button with the lowest tabindex, falling back to the first defined if nothing else is defined. There’s also some complications regarding whether the submits have a non-default value attribute IIRC.
The point to take away is that there is no defined standard for what happens here and it’s entirely at the whim of the browser – so as far as possible in whatever you’re doing, try to avoid relying on any particular behaviour. If you really must know, you can probably find out the behaviour of the various browser versions, but when I investigated this a while back there were some quite convoluted conditions (which of course are subject to change with new browser versions) and I’d advise you to avoid it if possible!