If the barbarian has no healing surges, she cannot use Second Wind. It requires her to spend one, which she does not have. And since the "Regain Hit Points" clause of Healing and Dying specifies that you have none left, this rule would never apply.
Second Wind[DDI] states:
You can dig into your resolve and endurance to find an extra burst of vitality. In game terms, you spend a healing surge to regain some of your lost hit points, and you focus on defending yourself. Unless otherwise noted in the statistics block of a monster or a nonplayer character, this action is available only to player characters.
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The errata'd version of healer's lore only works if the target character has spent a surge.
It does not work for powers that act as if they'd spent a surge.
So your second bullet is the correct interpretation. This was the purpose of the errata. Let's dig into this.
The new feature reads:
lets a creature spend a healing surge to regain hitpoints
So what does "spend a healing surge" actually mean?
RC 258 has some details here:
Most healing requires a character to spend a healing surge. When a character spends a healing surge, he or she regains lost hit points and adds them to his or her current hit point total.
Once per encounter, a character can use the second wind action to spend a healing surge and regain hit points.
A character can spend a limited number of healing surges per day. When the character takes an extended rest, his or her number of healing surges is replenished.
After a short rest, the character can spend as many healing surges as desired outside combat.
Some powers allow a character to regain hit points as if he or she had spent a healing surge. When a character receives such healing, he or she doesn’t actually spend a healing surge. (emphasis mine)
So spending a surge is only when a surge is spent for healing.
The language of Healer's lore is explicit. It only triggers when a surge is spent, not when powers that function "as if a surge was spent".
Just to clarify that this is the intent, Greg from the Wizards team says it explicitly:
You don't gain the bonus to healing in powers that say "as if you had spent a healing surge."
One last note, since this is obviously not nearly as strong as it was pre-errata you might consider (if Dragon content is allowed in your game), the Battle Cleric's Lore option from Dragon 400, I haven't built a cleric with Healer's Lore in ages, as Battle Cleric's Lore addresses a gaping AC hole for Clerics. Or if you're building a solely PHB Cleric, you might argue for your DM to allow you to use the pre-Errata Healer's Lore as the PHB didn't have powers like Astral Seal that abuse it to great effect.
Best Answer
They can't, but it's probably a mistake
You're correct that a level 1 character has 1 hit die and since Healing Surge lets them use half, rounded down, it doesn't do anything for them.
That said, there was an identical issue with recovering hit dice that was corrected once it was pointed out. I expect that the intent is for there to be a minimum of 1 hit die for healing surge as well.