A 'subject' (he starts, feeling the mischievous impulse to confuse matters encroach) is the thing that does the doing. Almost certainly a noun (see, we have already eliminated your 'pronoun, adjective, preposition classification of words in my first sentence. When you are classifying words you… look, if you saw a playground full of children you could split them into groups by age, sex {knowing you}, house colours, height – a thousand ways to separate them {including the long-advised crowbar}. We do the same for words, which are generally more disciplined), it is the reference point of the sentence. In the simplified sentence "Spot ran", Spot is the subject, running is the verb, the action. One level of complexity, "Joe chased Spot". Now, Joe has replaced Spot as the subject while Spot himself, while as far as we know in no way objectionable, has become the object of the sentence, the thing acted upon by the subject, and the verb, the action has become 'to chase' (the 'action upon' would have been more evident if the sentence had been "Joe strangled Spot", but that wouldn't have been nice).
"Joe chased Spot down the garden." has changed nothing. The garden, while a noun, hasn't entered the structure, it's just sitting there, probably waiting for fairies to colonise its bottom. But (starting a sentence with a conjunction, which is frowned upon) if "Joe chased Spot down the garden, and they both fell into the duck pond.", we suddenly have a second verb ('fell', from 'to fall' which could have been a noun or an adjective with the same spelling, but is definitely an action in this case and thus verbal, while the 'duck', which had the same three options is preening itself adjectivally and… I'm making this too complicated, aren't I?), another subject (both) and nearly an object. We have run on, and got wet and muddy, and still the sentence shows nowhere near the level of complexity of most of mine (ignoring parenthetical interventions, which is probably the best way to treat them).
But that sentence contains the elements of two complete sentences, and could so be expressed "Joe chased Spot down the garden. They both fell into the duck pond. The ducks disapproved loudly at this invasion of their tranquillity." Oops. Or welded together with the addition of a conjunction (in this case 'and', but could have been 'but' for a contrast, or 'or', or a word like 'while' – Glisterspeck was somewhat simplifying the task) to make a single, larger concept. Which can have a comma in it "Joe chased Spot down the garden, and they both fell into the duck pond.", or it can be left out "Joe chased Spot down the garden and they both fell into the duck pond.", depending on whether you would take a breath at the junction, and thus subtly chances the emphasis in the sentence.
Now, for "See Spot run." the true subject is the unstated second person to whom the command is being given, the 'you' that should 'see', while 'Spot' is responsible for a second verb (to run) which makes that structure remarkably complicated for so few words. See?
He crouched down and as she returned to her task, he snapped a few shots. Satisfied, he stowed the camera away again and moved off through the trees, whistling for the dog to follow.
Your movement of the comma was correct, and suggests you should be listening to your instincts rather than analytic considerations ('specially mine). You put in the double commas, like a set of brackets (parentheses) but less so, round the subordinate clause "as she returned to her task", which could well have been taken out in its entirety leaving a grammatically complete (albeit less effective) sentence.
