This is a tricky situation to discuss and may sound like a matter of semantics...but a commoner or Hon. who marries into the a higher titled family is never entitled to use her own Christian/given name in conjunction with her husband's title e.g :- Lady Randolph Churchill was never formally known as Lady Jennie Churchill....the examples are too manifold to give here. But if a woman of the rank of Lady (e.g the daughter of an Earl, Marquess or Duke marries a man of equal rank she is entitled to use her own given name and her husband's surname in conjunction with her birthright title). The same principle applies to women who marry Princes of equal or unequal rank to themselves.
Princess Alice, Duchess of Gloucester never had the 'right' to call herself as such until after 1975 or 1976, where after she only used the title by 'courtesy' strictly speaking, I am afraid I can not remember precisely the year the Royal decree/letters warrant were issued, anyway.... she was technically and officially for the sake of debate in fact HRH Princess Henry, Duchess of Gloucester from the moment she married until the day she died....the change to Princess Alice's style only came about because she apparently loathed being known as HRH the 'Dowager' Duchess of Gloucester officially and in general terms, and preferred to find an alternative! The reason she had to find and alternative form of style and address was because her daughter in law became HRH the Duchess of Gloucester when Henry died in 1974....to have two HRH Duchesses of Gloucester would have lead to great confusion thus the use of Dowager to differentiate the widowed from the extant wife of the in situ Duke became an essential necessity! God this sounds complicated but it does I hope make sense?
However, this rule of thumb and tradition does applies across the spectrum... all living widows of deceased title holders are differentiated in style from the de facto wife of the living title holder as well as one another! At one stage, there were four widows(including an ex-wife) of Dukes of Marlborough and a wife of the incumbent Duke alive at the same time....the 'Dowager' adage was used by one and the rest used their Christians names followed by their dead/former husbands' title....hence Jane, Duchess of Marlborough,Frances Duchess of Marlborough, Bertha Duchess of Marlborough (officially she chose to be know as Bertha, Marchioness of Blandford), Lillian, Duchess of Marlborough (she was the de facto Dowager Duchess) and the Duchess of Marlborough (Consuelo), her name was never used formally as she was 'the' Duchess! Anyway... this is just an example of how complicated titles and styles may appear.........going back to Alice and.......
Princess Marina having been born an HRH had a choice and could have adopted her husband's title and name becoming HRH Princess George, Duchhess of Kent (just like Chelsy Jones from down the road marry Harry Smith and becoming Mrs Harry Smith) but because Marina was a Princess by birth, she was entitled to use her christian name in conjunction with her own title by 'right' and her husband's titles as she pleased. Generally women do or at least did until quite recently adopt their husband's name in full..being known say,as an example as Mrs Harry Smith and in the event of her becoming a widow she would then become known as Mrs Chelsy Smith! Until the Duke of Kent's death I believe Marina chose to be be known officially simply as HRH the Duchess of Kent and that she continued to be known as such until her elder son married Katharine Worsley (who at the moment of her marriage became HRH Princess Edward, Duchess of Kent known generally as HRH the Duchess of Kent)... thereafter Marina became known officially as HRH Princess Marina, Duchess of Kent to differentiate.....even though she had always been a de facto Princess Marina regardless of this fact.
With regards to Alice Gloucester......she was never known as or by the style rightfully or officially of Princess Alice until she and the Queen agreed to it as a 'courtesy', before then, she was always officially HRH Princess Henry, Duchess of Gloucester and generally known as HRH the Duchess of Gloucester, HRH the Dowager Duchess of Gloucester or 'unofficially and incorrectly) HRH Princess Alice, Dichess of Gloucester.
I guess that this confusion ultimately stems from everyone calling Diana - Princess Diana - a title which was neither officially correct or appropriate technically!
Goodness I hope this makes some semblance of sense!?!?!