"I think she does understand my intentions when I start putting her toys away."
Wow, as someone else posted -- please dial the expectations, and your belief that she is at all focused on your adult intentions, way down. Please consider getting some good books on child development and learning more about stages and ages -- "what to expect in the early toddler years" kinds of guidance.
At her age, every single thing in her world and her mind is immediate. She makes no connection at all between pulling the book off the shelf and putting it back onto the shelf. If, later, she puts the book on a shelf (not necessarily where it "belongs"), it's because she thought it would be fun to pick up that object and do that action with it, not because she has any thought yet of "tidying up" after her earlier action with that object. Can you see the difference between what you're expecting and intending, and how her very young mind works right now? This is why children this age make what adults see as "messes" and are not in the least affected by those messes and have no inclination to "clean up." They don't see it as mess and don't have a concept that these toys or books are out of place. Yes, you will have to pick things up for a long time to come.
What you can start to do now that is age-appropriate is to make what you think of as clean-up into what she will -- over time, she's still very young -- come to see as a game.
You are already part of the way there because you're "encouraging her to drop toys into a bag/box" and you say you've used the clean-up song. It's good that you have her hand you toys and you put them away yourself. Do you use the song every time, so she associates it with the game of putting things away? That would be helpful.
Do you want her to hand you item after item and "clean up until it's done," and find yourself frustrated when she loses interest and wanders off after two or three items? I'm betting that's one thing that happens here. Do have her hand you things but don't expect her to do it more than a couple of times; praise her a lot for doing so and at the same time name the item several times, say, "The red dog goes into the blue toy box!" and so on -- this helps her with verbal skills and much more but do not expect it to get the cleanup job done. At her age, she will hand you a few things and you need to do the rest. As she gets older you can start with directions such as: "Can you put the red dog into the blue box?" Then you'll move on to "Let's have a race! Can you put the red dog into the blue box before mommy can put this elephant in the box?" But don't try this now; you need to give her time and build up this game. And still don't expect her at two or three to do lots of items all in one go.
One other thing -- As she gets older, even when she's in elementary school, please resist saying "Clean up your room." That is an overwhelming statement for most children that age. They don't know where to begin and it seems enormous, so they do nothing and their parents get so frustrated. Give very specific tasks: "Please pick up your clothes off the floor and put them into the hamper." "Please put Elephant and Red Dog into the toy box." There are a lot of posts on here from moms who don't get why "Clean up your room" doesn't get results. It's because, for younger kids, it's just too much and too vague. Again, look into what's age-appropriate and praise your child for doing what's age-appropriate rather than expecting behaviors that aren't quite there for her yet. It takes a long time.