Short answer: yes, your citation from the PHB supports character initiative or action as the trigger from non-combat to combat.

It's not necessarily "another skill" that can set up surprise, but player actions, preparation, and decision.

Surprising Foes. If the adventurers encounter a hostile creature or group, the DM determines whether the adventurers or their foes might be surprised when combat erupts. (p. 65, Basic Rules)
Any character or monster that doesn’t notice a threat is surprised at the start of the encounter. (p. 69 Basic Rules)

While the second point is under the "Combat" rules, the surprised condition means that for some or all members of one side, combat actions start later than the other side since actions based on surprise (or lack thereof) are taken into account before any initiative order turn based actions and reactions play out: it complicates the implementation of "roll for initiative!"

If you’re surprised, you can’t move or take an action on your first turn of the combat, and you can’t take a reaction until that turn ends. A member of a group can be surprised even if the other members aren’t.

Two points worth noting:

• Surprise can be determined to affect a whole side, or individuals.

• From a rules perspective, surprise is explicitly linked to combat in terms of its significance, which is addressed in detail here.

• If a party is surprised, but no combat ensues, a lot of the detailed treatment of surprise is rendered irrelevant.

Discussion

Don't let the rules be an obstacle to play

In an attempt to "follow the rules as strictly as possible" in the perspective of "Rules as Written" it is well to also embrace the two other manifestations of the rules in this edition's design as expressed by Jeremy Crawford here.

RAI ... “rules as intended.” This approach is all about what the designers meant when they wrote something. In a perfect world, RAW and RAI align perfectly, but sometimes the words on the page don’t succeed at communicating the designers’ intent. Or perhaps the words succeed with one group of players but fail with another. When I write about the RAI interpretation of a rule, I’ll be pulling back the curtain and letting you know what the D&D team meant when we wrote a certain rule.

RAF. Regardless of what’s on the page or what the de-signers intended, D&D is meant to be fun, and the DM is the ringmaster at each game table. The best DMs shape the game on the fly to bring the most delight to his or her players. Such DMs aim for RAF, “rules as fun.”

The transition isn't all about "roll for initiative!"

The transition from non-combat to combat in an encounter is based on a trigger, an action, or a decision. Your question gets at the heart of that transition, which is often something other than the DM saying "roll for initiative." By setting up the trigger condition, and letting it develop that way, the DM empowers players and ensures their agency (if they do a good job of setting it up and describing the set up/plan, etc). This gets at the fundamental structure of play.

How the game is played:

(Basic Rules, Page 3)

1. The DM describes the environment
2. The players describe what they want to do.
3. The DM narrates the results

Note on point 3: the DM determines results (like whether surprise happens or not), which includes taking into account what effort the players are exerting to set up that condition.

When there is a chance for failure, or when success or failure is an interesting outcome (such as "did we achieve surprise or not?" in your scenario) then any die roll is based on DM Judgment. That is also what the rules spell out. The DM's role includes the requirement to make rulings. The rules in D&D 5th edition are not computer code. They are not a string of on/off switches. They include judgment, which is a human aspect, and which opens the door for a DM to determine "you make that attack with advantage because ..." or "you make that check with disadvantage because ..." and so on.

About terms:

• DM Fiat far too frequently attracts a pejorative connotation, which is unfortunate. DM judgment, (and a DM choosing not to metagame but instead role-play the NPCs/Monsters) is what makes or breaks a game.

• It isn't a house rule for a DM to act like a DM.

  Then the DM determines the results of the adventurers’ actions and narrates what they experience. Because the DM can improvise to react to anything the players attempt, D&D is infinitely flexible, and each adventure can be exciting and unexpected. (p. 2, Basic Rules)

DM judgment is part of the rules. (The infamous rule zero). If a DM doesn't apply judgment, then that role isn't being fulfilled.

Can we do this?

Absolutely, yes. The players characters attempting to lull the NPC's into not expecting an attack, and then suddenly attacking, is the essence of a surprise attack or an ambush, and is their input to the game flow (step 2) once the DM's description of the setting (set 1) is fulfilled.

Which opposed ability checks, if any, are necessary are a function of steps one and two: the DM describes the environment, and the players describe what they are attempting.

If successful, they'll surprise the guards. If not, oops, the jig is up!

Yes, with a readied action.

If B readied an action to move away when A entered his space–a perceivable condition–they would effectively swap places on A's initiative. This seems to be allowed by RAW, as neither player ends any movement in another's square, if the movements are resolved simultaneously. If movement is not resolved simultaneously, it should still be possible, with some fancy footwork, and a different trigger for B's Ready–that A begins to move.

Both options cost B's action on his own turn, however.