I was thinking about how we use the word ‘pebble’ to describe small rocks, and ‘boulder’ to describe very large ones. But where’s the cutoff? For me, I guess I’d describe a pebble as being a rock the size of M&Ms, but a boulder would be the size of my car.
Is there any formal definitions of these?
Well, my guess would have been that these terms are subjective, but looking up pebble or boulder in Wikipedia indicates that there’s a formal scale for such things.
I think of a pebble as fitting in your pocket, and a boulder as being deadly if it fell on top of you. But I’d go with Wikipedia.
I am a geologist and I endorse the Wentworth Scale mentioned in Wikipedia.
Technical definitions and common English usage seldom coincide for any argot. The Wentsworth scale makes any rock over a foot in diameter into a boulder. That’s not common English.
True, but in this case the OP asked for a formal definition.
Even that depends on the context. If I saw a one-foot-diameter rock, I’d probably just call it a large rock. But if I had to carry around several such large rocks for a small landscaping project, later that day I’d be complaining about all of the boulders I had to move.
Since this has been answered, I wanted to present the classic SNL skit, Tales of the Runaway Boulder. Sadly, I couldn’t find a clip… The last tale would illustrate this question nicely.
Heh 
My common English WAG would be that a boulder is big enough that one, maybe even two people are incapable of lifting it manually.
Now, if these things are falling down on me from up high, the size of a “boulder” would be a fair bit smaller.
How I’m understanding it is that, boulders and pebbles are not separate categories, but lie on a gradient. Contrast that with geological terminology, in which science tries to create distinct groupings. In the end, the difference between a pebble and a boulder is not empirical, but subjective. However, in science, one ought avoid subjective observations, so an arbitrary scale is made up to organize and categorize. Is my understanding correct?
I think so, but at the lower end (clay vs. silt vs. sand), there are some objective reasons for the divisions: different types of rock tend to naturally weather to different sizes, so clays and sands are usually chemically different, not just of different sizes.
A minimally-sized (256 mm) roughly spherical boulder of granite would weigh ~24 kg or 50 lbs… Most adults could lift that easily, but you would know you’re lifting something substantial (remember when you’re comparing it to your bowling ball or luggage that it has no handles or grips).
I got the impression - maybe incorrectly - that in America the word “rocks” can be used for what could only be called “stones” on this side of the Atlantic. An extreme example is in “A Widow For One Year” by John Irving, where a character says “She threw rocks at you?”, when what was actually thrown was gravel or pebbles from a driveway.
For me a pebble would have to be around the size of a cherry or less. A boulder? That’s big - I would agree that a boulder would be much too big to lift.
Where is meteor on this scale? 
FWIW, a boulder is formally a rock larger than 10 inches diameter. A pebble ranges from 0.16 inches to 2.5 inches, on the scale.
Right; on this side of the pond, “stone” and “rock” (and their plurals) are pretty much synonymous. There are shades of meaning, but I don’t think there’s any context where one would be regarded as correct and the other incorrect. Roughly speaking, if it’s worked, it’s more likely to be referred to as “stone”, and if it’s natural, it’s more likely to be “rock”.
The Wentworth scale seems about right to me - clearly in a casual context, it’s all down to local definition - for me, it would be something like:
Pebble: something small enough to conceal in my closed hand, crucially, worn smooth and rounded, or it’s not a pebble.
Stone: Something small enough to be thrown.
Boulder: Something big enough that I would struggle to carry, or would find altogether impossible to lift. Compact shape (or else it’s a slab or something else)
Rock: Something immovably large - part of the landscape.
Well, other than that there are two meanings of “rock” that “stone” is not synonymous for: 1) the actual matrix made up by one or more specific minerals. Granite includes quartz, feldspar, and mica; dolostone includes calcite and dolomite. Think of it as analogous to chemistry: the minerals are the compounds (and occasionally native elements); the rocks are the mixtures in which they are normally found. 2) Consolidated solids, firmly attached to the planet, constitute rock, as in bedrock, caprocks over underground reservoirs, etc.