It's both, not one or the other. It started in Gethsemane and was completed on Golgotha.
We do not know, we cannot tell, no mortal mind can conceive the full
import of what Christ did in Gethsemane.
We know he sweat great gouts of blood from every pore as he drained
the dregs of that bitter cup his Father had given him.
We know he suffered, both body and spirit, more than it is possible
for man to suffer, except it be unto death.
We know that in some way, incomprehensible to us, his suffering
satisfied the demands of justice, ransomed penitent souls from the
pains and penalties of sin, and made mercy available to those who
believe in his holy name.
We know that he lay prostrate upon the ground as the pains and agonies
of an infinite burden caused him to tremble and would that he might
not drink the bitter cup.
We know that an angel came from the courts of glory to strengthen him
in his ordeal, and we suppose it was mighty Michael, who foremost fell
that mortal man might be.
As near as we can judge, these infinite agonies—this suffering beyond
compare—continued for some three or four hours.
Finally, on a hill called Calvary—again, it was outside Jerusalem’s
walls—while helpless disciples looked on and felt the agonies of near
death in their own bodies, the Roman soldiers laid him upon the cross.
With great mallets they drove spikes of iron through his feet and
hands and wrists. Truly he was wounded for our transgressions and
bruised for our iniquities.
Then Alma 7:11-12 in the Book of Mormon:
11 And he shall go forth, suffering pains and afflictions and
temptations of every kind; and this that the word might be fulfilled
which saith he will take upon him the pains and the sicknesses of his
12 And he will take upon him death, that he may loose the bands of
death which bind his people; and he will take upon him their
infirmities, that his bowels may be filled with mercy, according to
the flesh, that he may know according to the flesh how to succor his
people according to their infirmities.
The suffering in Gethsemane was an important part of taking upon Him the pains and sicknesses of His people. Calvary was vital so He could take upon Him death.
The Bible refers to this suffering in Luke 22:
39 And he came out, and went, as he was wont, to the mount of
Olives; and his disciples also followed him.
40 And when he was at the place, he said unto them, Pray that ye
enter not into temptation.
41 And he was withdrawn from them about a stone’s cast, and kneeled
down, and prayed,
42 Saying, Father, if thou be willing, remove this cup from me:
nevertheless not my will, but thine, be done.
43 And there appeared an angel unto him from heaven, strengthening
44 And being in an agony he prayed more earnestly: and his sweat was
as it were great drops of blood falling down to the ground.
Emphasis added, of course.
I'm still mobile but will get more Biblical references in a bit.
To answer your other question, this has been the teaching of the LDS Church since its organization in 1830.