PocketKNIFE

Given your closing/locking mechanism requirement, i.e. no liner lockers, the ISO standard answer is to get a Benchmade 535 Bugout, or a 532 Mini Bugout. Both knives are very svelte in terms of thickness, and the Mini variant should also be small enough not to intimidate too much.

Except.

The Bugout currently costs $180 and, fine knife though it may be, is an incredibly bad value for the money. (Likewise the other small Axis Benchmades, including the Freek and the Full/Partial Immunity.)

I would instead look for a cheaper crossbar locking knife that will inherently have the same action. The Benchmade "Axis" lock, with its patent now expired, is being copied by other manufacturers and called various things including the above, or Kershaw's "Duralock," and similar. I find that the Axis/crossbar lock arrangement automatically causes one to hold the knife in such a way that your fingers are out of the way of the blade when you unlock and close it.

Consider possibly the Boker Rockstub or at the outside, the Kershaw Monitor. The Rockstub is the smaller of the two. I would also pitch the Ganzo D727M, although this might have a blade that's too long (it is over 3").

Debarking wood is actually very hard on a knife's edge, so a traditional knife will require frequent resharpening if used primarily for this task. In light of that, as plan C I would suggest the Cold Steel Finn Wolf which has a Scandi grind that should hold up to woodland abuse pretty well. Is is a lockback folder, but the blade is longer than your requirement.

Another option is the Spyderco Ladybug, which is a lockback folder with a 1.9" blade.

Yet another option if a button locker is preferred could be the Tekut Pecker, which is one of the very few button locking knives I've ever met that I actually liked, and was not an automatic.

Most of the available decent options in your size class are unfortunately going to be liner lockers, and if she is not comfortable with those at all it's going to limit your selection severely. Resist the urge to buy a non-locking knife, especially a Swiss Army Knife, which despite their popularity are actually kind of crap.

Other things to avoid because they are likewise crap: Gerber (as you have observed), current Schrade knives, Smith & Wesson (both of the preceding now made by Taylor Cutery, who manufacture Chinese garbage), Camillus (now a Wal Mart house brand), and Buck. I'm going to catch flak for dissuading you from buying a Buck, I just know it. Their old knives are quality. Their recent knives are not.

Maybe a folding box cutter. The blades are very short and disposable, so you won't need to worry about sharpening them.

Spyderco Dragonfly

Spyderco Delcia

Both are fantastic knives for small hands

Maybe a CJRB Pyrite mini? The blade is just over 2" but otherwise suits your requirements.

Kershaw, Ken Onion Leek is a great knife for the money, and I think it perfectly fits your criteria.

Edit: I just saw your comment about the locking mechanism. In that case you're going to want a Benchmade Mini Griptilian, for their axis-lock, but it's $50 over your budget, instead of being $50 under. I'd suggest just teaching them how to use a thumb lock, since that is the most common type of lock on good knives.

Edit 2: if you're cool with it being a belt knife instead of a folding knife, then the Morakniv Carbon is an amazing knife considering it only costs $15.

https://www.buckknives.com/product/112-slim-pro-trx-knife/?sku=0112ORS3-B