I’m not sure that the priest was right to omit the truth that God is triune while praying for your mother. It may have been a nice sentiment, but it also very well could have been a sin against faith on the part of the priest.
I suppose rabbis may pray for the health of a Catholic, but I don’t think it should be done in front of her, or at least, it should not be expected that other Catholics join in such a prayer because there is at least a possibility for an implicit denial of the Holy Trinity (a Catholic cannot think, “Oh, it doesn’t matter how or in what way we worship God, and it doesn’t matter what we call Him”). Historically, the Church has been very strict about Catholics worshipping with non-Catholics (and vice versa). Private prayer between different Christians has been allowed, provided there is nothing heretical (i.e. erroneous doctrine) in the prayers. Prayer between Catholics and Jews presents a much bigger problem because Jews deny the divinity of Christ and the revelation of God in three Persons. We are to pray “through Christ our Lord” or “in the name of our Lord Jesus Christ,” so we cannot truly pray with Jews. Our differences in belief are greater than some people would like to admit (including our Catholic prelates).
The rabbi should pray privately for the Catholic. I hope my advice has been helpful.