According to NOAD, the definition of home in this context is given thus:
home (adv)
to or at the place where one lives
Technically, it is not ungrammatical to say The children are playing home to mean The children are playing at home, since at is implicit in the definition of the adverb home. However, this form is certainly not colloquial, and I daresay no native speaker would use this in any context, formal or informal.
Here are some examples in which it is all right to drop at (informal contexts):
- He's staying home today.
- When is she usually home?
- I missed your class because I was home sick yesterday.
In the following examples, it would be incorrect to include a preposition:
- When is mom getting home?
- I want to go home!
- Son, when are you coming home?
- Are you driving home today? I need a ride.
You may notice that home is generally used alone as an adverb of place (without at, from, etc) with movement verbs, e.g. come, go, leave, stay, be, drive, dash, move, etc. Verbs of other species usually require at, e.g. dance, play, eat, relax, and whatever else one does at home!
To my ear eat home sounds all right, and I may have heard or used it myself. This may be one of a few exceptions, but it is certainly not formal. If eat home is ever used, my hunch is that eat in (which means eat at home) is falling out of favor, or, as native speakers tend to do, at is dropped for convenience and it sounds good!
- We're eating home today. (Technically, it is eat in if at is not used)
Another observation to note is in the case of non-movement verbs, the preposition may be dropped (colloquially) when the action described can be performed at home as opposed to somewhere else. Examples are few and far between, though.
I was really thinking to do that.
You might have assumed I was thinking about doing this, but actually that is what I have in mind (the word really confirms that as the option being considered).
I really was thinking to do that.
You might have assumed I had no real plans, but actually I was really thinking of doing that (the word really confirms the intention to do something).
In both cases, the word really implies the speaker thinks his audience may be labouring under a misapprehension (about the specific plan, or the fact of having any serious plan at all).
Note that both OP's examples are generally considered informal/non-standard. Thinking to do is far less common than the "standard" forms thinking of doing (or about doing).
Best Answer
Both are correct; you would choose according to context.
The first suggests that the fall is the topic (i.e. you are explaining the circumstances of a fall, for example to a doctor) while the second suggests his play is the topic (you are describing his play in the field, and mention the fall as the next event in a sequence.)
The ambiguity of the sentence structure arises because "when" has slightly different meanings in the two contexts. In the first sentence, you use "when" to introduce the answer to a potential question:
When did he fall down?
He fell down when he was playing in the field.
In the second sentence, the word "when" is actually used in place of the now-uncommon "whereupon," meaning "at which time":
He was playing in the field, whereupon he fell down.
In speech however, the second sentence could convey the same meaning as the first, provided emphasis is place on the first clause, especially the gerund and noun playing and field. In this case the sentence-reversal is used to emphasise the circumstance of the fall by bringing it to the beginning of the sentence.