Is correct. Coins have 'DEI GRA REG' which stands for 'Dei Gratia Regina' which in turn is latin for 'by the grace of God - Queen'. Old coins when a king was monarch had REX instead of REG. On smaller coins it might be contracted to D G REG.
FID DEF stands for Fidei Defensor. Latin for 'Defender of the Faith' the title that Henry VIII took on when he established the Church of England.
Close, but it's actually more ironic than that. Henry VIII was given the title of "fid Def" by the Pope, Leo X, after Henry wrote a book,
assertio septem sacramentorum, in which he asserted the sanctity of marriage and the supremacy of the Pope's authority. Henry somewhat famously changed his stance on both subjects but retained the title as he felt it leant weight to his role as head of his own new Church.
Henry was excommunicated and the title was formally rescinded by Pope Paul III, but an Act of Parliament restored the appellation and it remains to this day.
Prince Charles has confirmed that he would instead adopt the title "Defender of Faiths" and promote freedom of worship, when he acedes the throne.