I'm wondering if it's just a majority needed from those present (provided there are 51 senators present in total), or if it's a majority in terms of total senate seats (51)? Is this true even for the Supreme Court nominations?
Technically speaking, the filibuster is a dead letter rule. The nuclear option is invokable to change the rules any time the majority sees fit. Thus we have seen its use to limit the filibuster in
- General judicial nominations and Presidential Cabinet positions (2013, Democrat majority)
- Supreme Court nominations (2017, Republican majority)
There's no reason it cannot be invoked for general legislative matters.
To your question, the Senate Reference on Rules and Procedures state
Article I, section 5 of the Constitution requires that a quorum (51 senators) be present for the Senate to conduct business. Often, fewer than 51 senators are present on the floor, but the Senate presumes a quorum unless a roll call vote or quorum call suggests otherwise.
Since the majority does not need the opposition to establish a quorum, nor to pass the vote, you should not need any more than 51 Senators to consent to a nomination.
First some background: approval of Presidential nominees (or any other matter) always required only simple majority vote. The trick was that the Senate rules required three-fifths of all Senators to affirmatively vote to end debate. Thus if any Senator were to filibuster a nominee, 60 Senators (and no less assuming that all Senators are chosen and sworn) to end a filibuster. In that case less than 60 Senators could never end debate and thus a filibuster with enough support would defeat a nomination.
According to https://www.senate.gov/CRSpubs/be873e40-a966-4feb-9d72-cf23a93cbe46.pdf only majority vote is required to invoke cloture on all nominees except for Supreme Court Nominees (Until April 2017). I wanted to check the record to confirm and further supplement this answer.
According to page 5 and 6 of the Senate Congressional Record on November 21, 2013 (see https://www.congress.gov/crec/2013/11/21/CREC-2013-11-21-senate.pdf, pages referenced numbered S8417 and S8418), Senate Majority Leader Reid made a point of order that the cloture motion requires a majority vote. The President Pro Tempore (acting as the Chair of the Senate) ruled that this point of order is not sustained. In response, Senator Reid appealed the decision of the chair. The appeal was carried.
Afterwards, Senate Minority Leader McConnell, made a point of order that the appeal required 2/3rds vote and not majority as it effectively changed the rules of the Senate. The chair indicated that his point of order was not sustained and Senator McConnell in turn appealed that ruling. It was not carried.
Thus the rules of the Senate as of November 21, 2013 indicate that three-fifths of the currently chosen and sworn are required to invoke cloture. However, the rules have been interpreted (funny things happen when the politicians who write the rules also need to interpret them) to only require majority vote to invoke cloture on all Presidential nominees except for Supreme Court nominees.
Now this was precedent of the Senate until April 6, 2017. On that day a motion to invoke cloture on a Supreme Court nomination was not carried (55-45 for those interested). After the motion to invoke cloture did not carry, Senate Majority Leader McConnell asked to reconsider the vote and as reconsideration requires only majority, this carried by a vote of 55-45. Afterwards, Senator McConnell made a point of order that in fact the 2013 appeal covered all nominees, including Supreme Court nominations (note: it explicitly did not). The chair did not sustain this point and Senator McConnell appealed the decision of the chair. The appeal was carried by a vote of 52-48, (see pgs 7-8 here: https://www.congress.gov/crec/2017/04/04/CREC-2017-04-04-bk3.pdf S2390 is the exact numbering on the page).
As of this date all executive nominations require majority vote to invoke cloture. Note the appeals never indicated that cloture need be invoked by absolute majority, only simple majority. As a result, if the Senate has quorum, it only takes majority vote to invoke cloture (assuming all other prerequisites to invoke cloture are met). Quorum is defined as majority of the Senators. Thus at least 51 Senators need to be present (which essentially means whenever you see a vote of 51-50, ie. when the Vice-President votes, the senators on the losing side could have taken the path of simply leaving the Senate without quorum, as the Vice-President does not count for quorum).
Assuming that at least 51 Senators are present and stay in the room and if more than half vote to invoke cloture (or without filibuster, unanimously agree to proceed to vote), then once again only majority is now required to approve the nomination. Thus 51 or more Senators present in the room is sufficient for the majority to confirm a nominee and overcome a filibuster.
Seeing as many Senators are eager to win re-election they are eager to put themselves on the record as supporting or opposing the President. As such this situation where less than an absolute majority approve a nominees is unlikely to occur.
In regards to other matters, the rule on cloture has not been “interpreted” in any unusual way for matters other than nominations. Thus since the nuclear option has not been invoked for those matters, three-fifths vote is still required to invoke cloture on those matters. However, without objection a simple majority of those present can approve things like laws and resolutions. The nuclear option can of course always be invoked on these matters and invoking the nuclear option only requires simple majority vote. Once that is done a simple majority vote can end debate and then proceed to a vote and once that is done, a simple majority vote on the actual matter carries the matter.