Why is the first lines of the Dhammapada twin verses sometimes translated using the word 'heart'?

Here is what I found:

Manopubbangama dhamma
manosettha manomaya
manasa ce padutthena

There's translation and commentary on the page above.

The associated story ("The Story of Thera Cakkhupala") suggests it might be 'intention'.

That's kind of compatible with the poetic use of heart: "he had a change of heart", "his heart wasn't in it", "he did it with all his heart", etc.

The root syllable in each line is "mano" and/or "mana": which can also be found here associated with "mental consciousness"; but see also this says that it's used for "mental action" i.e. kamma.

My guess is that:

• Pali is constructed using short syllables like 'man'
• The syllables have an approximate meaning
• The suffix changes depending on grammar
• The syllables are aggregated into larger compound words and/or used with other words in a sentence
• There may be a basic or root meaning (perhaps from the root's etymology) but the specific/precise meaning (of a syllable) depends on context
• The original specific meaning is non-trivial for Pali scholars to render using single English words (for example the several words used to translate dukkha)
• You might find it useful to know at least the approximate/root meaning
• You might also find it useful to know that the meaning is a bit approximate!

An example of one word in English which has many specific meanings is, just to give an example, "run": you can run a race, run a horse, run an engine, run a bath, run a business, let things run their course, run away, run the printing presses, run for President, etc. as described in a dictionary.

"What is the original Pali phrasing?"

Verse I

Manopubbangamā dhammā manosetthā manomayā,
manasā che padutthena bhāsati vā karoti vā
tato nan dukkha manveti chakkan va vahato padan.

Verse II

Manopubbangamā dhammā manosetthā manomayā,
manasā che pasannena bhāsati vā karoti vā
tato nan sukhamanveti chāyā va anapāyinī.

"and which words are able to be translated freely between mind and heart?"

I'm not sure (see @yuttadhammo's answer), but Thanissaro is using "heart" to translate manas.

"What is the overall sense of the Pali words used i.e. what are the nuances or subtleties to the original Pali?"

Sue Hamilton [1] discusses these passages:

In the Dhammapada however, both terms [manas and dhammā] are being used in their generic sense [not as sense and sense object, as elsewhere in the pali canon]: manas is being used as 'mind' in general, and dhammā is being used as it is in the tilakkhana [2] formula [...].

When these terms are used generically, the precise meaning of the passage has to be determined from context. The [first line of both verses] literally means something like: "Phenomena are the result of mind (or are preceeded by mind), have mind as their best, are mind-made". In isolation, this sentence might be construed as positing an idealistic ontology, that the phenomena which comprise the world as we know it, dhammā, are nothing but the mind: the external world is magically created by the mind and consist of mind. There is an alternative way of interpreting this line, however, which is, in my opinion, the accurate one.

Then she offers a full translation of the rest of the passage. Excluding the first lines:

"If a man speaks or acts with a wicked mind, sorrow follows him as wheel follows the foot of a draught [animal]...if a man speaks or acts with a pure mind, happiness follows him as a shadow always follows [him]."

Then she points out that:

The point of these sentences is that one reaps the consequences of one's actions [...]. This corresponds to the Buddha's definition of karma [volition] [...]. The whole of the first chapter in Dhammapada is concerned with this teaching: that one reaps as one sows, and that sowing is qualitatively determined by intention.

With the above in mind, and in the same vein as @ChrisW answer , "heart" could've been a choice for someone's inclination. Perhaps to avoid giving the impression of an ontological statement.

Finally:

I mentioned in chapter IV that when used in a non-technical sense, 'mind' (whatever term is being used) indicates general mental activity. So the first lines of the two verses in fact have the following meaning:

dhammā are an individual's experiences -- everything, in fact, that is a part of the individual's life. And qualitatively those experiences follow from one's mental activity: manopubbangamā, it is mental activity which principally governs the nature of life: manosetthā, and it is in mental activity that what follows originates: manomaya.

To interpret this sentence ontologically is completely to ignore the context in which it is found and to divorce it from the subject matter of the entire chapter. To convey the meaning of manas in this context accurately, it is better to translate it as a verbal noun, denoting the activity or process of the mind: 'thinking'. In english, this gives us a less ambiguous sentence:

"The individual's experiences are preceded by thinking, have thinking as their best, originate in thinking".

It is the reification of manas as 'mind' which tends to mislead here

[1] Identity and Experience: The Constitution of the Human Being According to Early Buddhism, pg 140ff

[2] The Three marks of existence: "sabbe saṅkhāra aniccā. sabbe saṅkhāra dukkhā. sabbe dhammā anattā"