Retzeh VeHaChalitzenu

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Tannaim and Amoraim

Already in the Tosefta it says that one mentions Shabbat in Birkat HaMazon. In the Bavli it says that on Shabbat one should be "מתחיל בנחמה ומסיים בנחמה ואומר קדושת היום באמצע". In Vayikra Rabbah it mentions saying "רצה והחליצנו" on Shabbat.

Geonim and Rishonim

In the Geonim and Rishonim there are different opinions about what "מתחיל בנחמה ומסיים בנחמה" means, does one change the whole berakhah on Shabbat and what is considered נחמה and based on that have different נוסחאות:

In ספרד they were חותם with "מנחם ציון ובונה ירושלים" and were פותח with either:

"נחם" - Rif, אשכול, אבודרהם. This way both in the beginning and the end there was the שורש of נחם.
"רחם" - R"Y b. Yakar. According to this, ר"י בר יקר would probably explain that "רחם" is also considered נחמה and so is enough.¹

In אשכנז and צרפת they were חותם with "בונה ירושלים" and were פותח with either:
Any פתיחה or חתימה - The Rambam and שו"ע say that one can say either "נחם" or "רחם" and either "מנחם ציון ובונה ירושלים" or "בונה ירושלים" as long as it is the same as what one says on chol.