Here is the correct Kate Dixon in the 1901 census, living with the same Lavelle family as in 1911, in same location
https://census.nationalarchives.ie/reels/nai001070544/
Kate appears to have siblings John and Martin.
The Catherine Lavelle in the 1901 and 1911 census returns is nee Catherine Dixon, widow of Anthony Lavelle.
You've done some good work on this, please confirm if my summary - in reverse chronological order - is correct:
* My great-grandmother - Kate Dixon - dies in 1975 aged 89; this is supported by the grave and the death certificate. This puts her birth year as around 1886
* She gave birth to five children to John Murphy, all in Belmullet: most recent is Thomas (my maternal grandfather); eldest is Mary (later went by the name of Maureen, mother to the actor Richard LeParmentier)
* Kate is living with cousin Anthony Lavelle's family with John and infant daughter Mary in 1911. The household includes her aged aunt Catherine
* She is living with aunt Catherine's family in 1901, along with her two kid brothers Martin and John. Also in the household are aunt Rose and her husband Michael.
Can we assume the following:
* The Dixon children are orphaned, given the young age of John Dixon?
* The Lavelle family have a larger-than-average farm, and are in regular need of labour. Along comes outsider John Murphy of Cork and he seems like a good match for Kate?