etymonline refers to it as:
"record of proceedings," c.1710, perhaps from Latin minuta scriptura "rough notes," literally "small writing;" see minute (adj.). Minute "rough draft" is attested from c.1500.
From this, I would infer that minutes are called like that due to its conciseness (as in size) in comparison to what happened during the meeting. This does in no way mean that is is less detailed, but that only the most important parts are recorded.
For instance, businessdictionary defines minutes as:
Permanent, formal, and detailed (although not verbatim) record of business transacted, and resolutions adopted, at a firm's official meetings such as board of directors, manager's, and annual general meeting (AGM). Once written up (or typed) in a minute book and approved at the next meeting, the minutes are accepted as a true representation of the proceedings they record and can be used as prima facie evidence in legal matters.
For example, wikipedia mentions:
Where a tally is included, it is sufficient to record the number of people voting for and against a motion (or abstaining), but requests by participants to note their votes by name may be allowed.
As long as we know what happened in this vote, who voted is not relevant and is not recorded in the minutes, unless requested.