My initial concept was:
One should finish one's meal.
When a user logs in to his/her email...
A person bought a cake for his/her work.
However, in the recent (8-10) years, I have been seeing more and more references like these:
One should finish their meal.
When a user logs in to their email...
A person bought a cake for their work.
Which of the two approaches above is correct? Do they have to do something with the difference between British and American English?
Some of this could be covered in the answers to another question, like Is there a correct gender-neutral singular pronoun ("his" vs. "her" vs. "their")?.
English (or American English, at least) is trying to deal with this idea of a sentient neuter third-person pronoun set. A person generally dislikes being called "it", so we try to extend that courtesy to others. Unfortunately, we don't have a standard.
Using "one" sounds stiff: appropriate under the right conditions, but almost as bad as "it" otherwise. Using "he/she", "his/her", etc., is uncomfortable if used too often, but can sound ok with the right delivery once or twice. Use of "they" is the most comfortable, though it tends to mean modifying the verb to agree with the third-person plural. There have been suggested neologisms, such as "zey", but these haven't caught on yet.
Until a standard is agreed upon and school-mastered into the language, this will probably remain an area of debate. I've heard that losing a war is one of the most common precursors to the school-mastering of a language.
