RecursiveDocumentSplitter split_idx_start is off by the trailing separator with split_unit="word" and split_overlap > 0

Describe the bug With split_unit="word" and split_overlap > 0, RecursiveDocumentSplitter sets split_idx_start one character (the length of the previous chunk's trailing separator) too far for every chunk after the first, and the _split_overlap ranges inherit the same shift. The repro from #11710 still fails on main (f1e61b5).

Cause, in _run_one: the position advances by len(chunk) - len(overlap_str), but in word mode _get_overlap rebuilds the overlap as " ".join(last_words), which has no trailing whitespace, while in the chunk those words are followed by the separator (" " with the default separators). So the subtraction leaves the position one separator past where the next chunk actually starts.

Error message None, the metadata is silently wrong.

Expected behavior document.content[split_idx_start : split_idx_start + len(chunk.content)] == chunk.content for every chunk (as the existing text.index(...) tests assert for the other modes), and the _split_overlap ranges of consecutive chunks point at the same text.

Additional context

from haystack import Document
from haystack.components.preprocessors import RecursiveDocumentSplitter

text = "This is sentence one. This is sentence two. This is sentence three. This is sentence four."
splitter = RecursiveDocumentSplitter(split_length=4, split_overlap=1, split_unit="word")
splitter.warm_up()
for doc in splitter.run(documents=[Document(content=text)])["documents"]:
    start = doc.meta["split_idx_start"]
    print(start, text.index(doc.content), repr(doc.content))
# 0 0 'This is sentence one. '
# 18 17 'one. This is sentence'
# 31 30 'sentence two. This is'
# 50 49 'is sentence three. This'
# 69 68 'This is sentence four.'

Same with separators=[" "] explicitly. split_unit="char" is correct.

To Reproduce Run the snippet above.

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System:

• OS: Windows 11
• Haystack version (commit or version number): main at f1e61b5
• Python: 3.12

I have a fix with regression tests and will open a PR. Found by property-testing the splitters; report and fix prepared with an AI assistant (Claude Code) and reviewed by me.