Increasing the skills is part of the significant milestone Benefit.
I think I look at it a bit differently:
* A minor allows you to change your skill pool (increase a skill at the cost of an existing skill).
* A significant milestone allows you to increase your skill pool (increase a skill without taking away from an existing skill), and also perform a minor benefit (possibly including an additional change to your skill pool).
Given that, I think it's within the spirit of the rules to perform the change the player wanted in one minor plus one significant milestone, as follows:
Start:
Lore 5,
Guns 4, Conviction 4,
Discipline 3, Endurance 3,
Resources 2, Rapport 2, Athletics 2,
Presence 1, Scholarship 1, Stealth 1, Alertness 1, Contacts 1
Minor Milestone(use the minor benefit to swap a point of Contacts for a (banked) point designated for use with Athletics):
Lore 5,
Guns 4, Conviction 4,
Discipline 3, Endurance 3,
Resources 2, Rapport 2, Athletics 2(3),
Presence 1, Scholarship 1, Stealth 1, Alertness 1
Note: At this point, actually increasing Athletics to 3 isn't legal, so it counts as Athletics 2 for now (the phantom third point is 'in the bank').
Significant Milestone(use the 'bonus' minor benefit to swap a point of Stealth for a (banked) point designated for use with Discipline):
Lore 5,
Guns 4, Conviction 4,
Discipline 3(4), Endurance 3,
Resources 2, Rapport 2, Athletics 2(3),
Presence 1, Scholarship 1, Alertness 1
(apply the significant benefit to gain a (banked) skill point, designated for use with Conviction):
Lore 5,
Guns 4, Conviction 4(5),
Discipline 3(4), Endurance 3,
Resources 2, Rapport 2, Athletics 2(3),
Presence 1, Scholarship 1, Alertness 1
Note: At this point, the three banked skill points can take effect while keeping the skill columns legal:
Lore 5, Conviction 5,
Guns 4, Discipline 4,
Endurance 3, Athletics 3,
Resources 2, Rapport 2,
Presence 1, Scholarship 1, Alertness 1
This could also be done with three minor milestones (which would require trading in the Alertness 1 to provide the third skill point), or potentially during a major milestone (which allows for skill reconfiguration if appropriate).
I think this is a fair way of applying the rules within the spirit of the rules. You are only increasing skills by a single point per benefit. Note that the skill points aren't really 'banked' in the sense of becoming unallocated; instead they are shifted immediately to their new skill, though you are delaying the use of those skills until they legally fit into your skill columns.
But, as always, play it the way that makes the most sense to you at your table.