Learn English – Carry at the back or carry on the back

In and at and on are notoriously difficult for learners to grasp and native speakers to explain.

These are called prepositions of place.

at conceives of a place or location as a point. Like an X on a map. You are at a particular place and not some other place. If there is an inside and outside to the place, at does not indicate whether you are inside or outside.

I'm at the bus stop.
I'm at the corner of 5th Avenue and Elm Street. =conceive the corner as the point of intersection.
I'm at the southeast entrance of the parking lot.
I'm at the park. (park is conceived as a point in space)
I'm at the parking lot. (conceived of as a point in space)
I'm at the shopping mall.
I'm at the south concession stand of Section 111 of the ballpark.
I'm at the swimming pool. (somewhere near the swimming pool, not in the swimming pool!)
I'm at the elevator. (as in at the elevator entrance)

Sometimes these expressions become fixed phrases, such as in
at school
at home
at work
at the office

In conceives of a place or location as containment or enclosure, often as inside or within.

I'm in the store. =inside the store.
I'm in the park. =within the park.
I'm in the parking lot. =within the parking lot.
I'm in in the garden.
I'm in the city.
I'm in the world.
I'm in the corner of my room. =confined in the corner
I'm in the swimming pool. =within the water.
I'm in the chair. =enclosed by the chair.
I'm in the car.=within
I'm in the bus/train/airplane/boat/trolley/streetcar=confined within or inside.
I'm in first class (section of the aircraft)! =confined in or within the first class section or compartment.

On conceives of a place as on a surface or in contact with a surface

I'm on the chair. =the surface of the chair, including the seat or the arm or the back.
I'm on the corner of 5th Avenue and Elm Street. =conceive the corner as a surface area.
Im on the world (and I want to get off) =conceive the world as a surface area.
I'm on the school =on top of the school roof, for example.
I'm on the automobile (car) =on top of the autombile (car), it's roof as a surface
Note well: I'm on the bus/train/airplane/boat/trolley/streetcar (conceive of being in contact with a surface; these methods of transport are large enough to have a walkway, which is the surface.)
The fly is on the wall =in contact with a surface.

Notice corner was used with all three; which preposition depends on how the speaker conceives of the corner (at the point of intersection of two lines, in a place that confines her, or on a surface area.)

Compare:
The bird is at the tree, as compared to at the bird feeder.
The bird is in the tree=within the tree branches, for example, or confined in the tree.
The bird is on the tree=on a branch, for example.

The bird is at the bird feeder, in the bird feeder (stuck inside, for example), on the bird feeder (on a surface, either on top or on a peg...

The boy is playing hockey on the street=street as a surface area.
The boy is playing hockey in the street=confined within the street.

Jill is at the stairway. (point in space)
Jill is on the stairway. (on a surface)
Jill is in the stairway. (within or confined)

Write your name in the top left corner of the page. (confined within the corner)
Now, print today's date at the top of the page (a point).
On the bottom of the page, draw an X.(on=on the surface of the page).
Draw a C in the middle of the page (confined in the middle).

Edit to add elevator:
We are at the elevator to mean at the entrance to the elevator.
We are in the elevator. We are inside the elevator.
We are on the elevator. (You are riding on the elevator, like on the bus. Here, the elevator floor is the surface area.)

Two people trying to find each other:

A. Where are you? (in some dialects, Where are you at?)
B. I'm at the zoological park.
A. Me too! Where at the park are you? Are you already in (inside, within the park)?
B. Not yet. I'm out on the parking lot (surface area) arguing with this other person.
A. What? What happened?
B. I was parking my car in the parking lot (within) when some clown hit my car. Where are you?
A. I was waiting at the 4th entrance like we agreed, but little Joey got bored, so we are already in (inside) the zoo.
B. Where in the zoo are you?
A. We were at the first taco stand; then we waited at the entrance of the Panda House for 20 minutes.
B. Cool! Are you in yet? (within/inside, the Panda House)
A. No. There is a delay. There is a worker on the Panda House, (on the surface of the top/roof).
B. Okay, I am now in line (fixed phrase) trying to get in (inside) the zoo.
A. I'll come meet you. At which entrance are you?
B. I'm at the southwest entrance.
A. Don't move. We're coming.
B. Too late. We're already in. (within, inside, confined in).
A. Where (at)?
B. I'm on the zoo train. (train conceived of as a surface). But these seats are so small, I sitting on the seat, not in the seat.
A. What do you mean on the seat?
B. Well, on the back of the seat (the back as a surface).
A. Are you (located/seated) at the back of the train? (a particular point ).
B. No, I never sit in back (fixed phrase, but means confined in back).
A. Guess what? We're on the ramp (surface) of the Lion Exhibit. Now we're in the exhibit (confined/inside).
B. What do you mean in the exhibit? You're really inside (=within, in) the exhibit, among the lions?!!
A.-----
B.Hello? Hello?...Well, I hope she is not in the lion!