Are the elect all those chosen by God to receive salvation?
Or are the elect only those chosen by God to serve a certain purpose, mission, ministry?
To understand this, you'll have to leave any path line of philosophical thought and stick strictly to what God's Word reveals.
In John 17 prior to Jesus being delivered up, He prayed to The Father. He spoke a lot about His elect chosen Apostles, and even revealed that they originally belonged to The Father, and The Father gave them to Jesus. He then said as He was not of the world, nor were they, and as He was 'sent', so He sent His Apostles, and they were justified.
The Lord Jesus then spoke of a second group, those who would believe based on their word (i.e., His sent Apostles preaching of The Gospel). Then Jesus prayed that both groups would together become one in Him and The Father, even as They are One.
It's like Lord Jesus said, ... "For many are called, but few are chosen." (Matthew 22:14)
That is based on the same idea of the two groups in John 17. Christ's sent ones (meaning of an apostle) were 'chosen'. But the other group that would believe by their word, are 'called' only. So what's the difference? And this is not philosophy, it is sticking to what God's Word reveals.
In Apostle Paul's Epistles, he often admonished the brethren to stand fast in the Faith, and don't fall away from Christ. In other words, Paul knew that with many of them it would be possible for them to be deceived, IF... they allowed it. That is the fate of those 'called' only. They MUST be tested.
But for the 'sent' ones, Christ's chosen ones, they have already been sanctified and justified, and He already 'owns' them. That's why He can divinely intervene in the lives of His chosen ones. If they stray, like Jonah tried to, He can direct them back on His Plan, and they cannot get out of their duty that God chosen them for. Apostle Paul especially is one of the prime Bible examples of this. Paul did not choose Christ, Paul was actually busy persecuting Christ's Church for his Pharisee brethren. Christ chose Paul, divinely intervened with Paul, removing any doubt of Who He was. That showed Christ's ownership of Paul.
When are those of us who are 'called' only known as the elect?
It is when we 'stay' in The Faith on Jesus Christ as The Saviour. We, Christ's sent ones, and Christ Himself, we all become 'one' in Him and The Father, His many-membered body, like Jesus prayed for in John 17.
This is why many get confused when reading Paul's Epistles, for in some places, like Romans 8, Paul talks about how we 'called' have been predestinated and justified and also glorified. That's talk as if those of us that have believed on Jesus could never... fall away. Yet in Hebrews 6 Paul talks about that very matter of some falling away; so does Apostle Peter in 2 Peter 2. And further, in 2 Thessalonians 2 about the end of this world, Apostle Paul forewarned about many falling away when that "man of sin" shows up to deceive the world.
In Matthew 24:24, Lord Jesus showed that the coming
pseudochristos ('a spurious Messiah' per Dr. Strong in his Strong's Exhaustive Concordance), would
IF it were possible, deceive even the very elect. That means it is impossible... for Christ's very elect to be deceived, because they are 'chosen', and He already owns them, and can show them the mysteries of the kingdom. This idea is also in Revelation 13:4-8 where we are shown about the whole... world worshiping the "dragon", EXCEPT those whose names were written in the Book of Life from the foundation of the world. That's another example of it being impossible... for Christ's 'chosen' elect to be deceived.
Likewise in the future, after Christ's future "thousand years" reign of Revelation 20, the unsaved of the nations are going to hear The Gospel during that thousand years (that idea hinted at in the Old Testament prophets). They will 'see' Christ Jesus then, as He will be King over all the earth, over all peoples and nations. But at the end... of that "thousand years", they MUST be tested, just as the 'called' will be tested in the coming "great tribulation" for this present world just prior to Christ's return. That is why at the end of the future "thousand years", Satan is loosed from his pit prison one final time, to go tempt the nations.