If the person writes under a "false name" then pseudonym is the answer.
If the person has chosen to call him or herself with one name only, such as Adele, Prince, Pelé or Sting, then the term is mononym. For example, “Her mononym is Shakira.”
A single name by which a person, thing etc is known. For example, Madonna (the pop musical artist)
Talking of Prince, when he changed his name to the so-called ‘love’ symbol 
, for a number of years he was known as "The Artist Formerly Known as Prince", "TAFKAP", or just "The Artist". Likewise anyone could rename him or herself and say:
I'm [new first name], formerly known as [old first name].
OR
I'm [new first name], but I used to be called [old first name].
If a person dislikes their first name (for whatever reason), they can just call themselves with a different name. Sometimes a family nickname sticks, and that person grows up with that name. Changing a name by deed poll might include a lengthy process depending on one's nationality.
The most common use is a name change through a deed of change of name (often referred to simply as a deed poll). Deeds poll are used for this purpose in countries including the United Kingdom, Ireland, New Zealand, Hong Kong and Singapore. In the UK, a deed poll can also be used to change a child's name, as long as everyone with parental responsibility for the child consents to it and the child does not object to it. The child's parents execute the deed poll on the child's behalf. In some other jurisdictions, a person may simply start using a new name without any formal legal process.
Wikipedia
Oxford Dictionaries:
pseudonym
(noun)
a fictitious name, especially one used by an author.
"I wrote under the pseudonym of Evelyn Hervey"
synonyms: pen name, assumed name, alias, professional name, sobriquet, stage name, nickname
Best Answer
(US English)
In Case 1, the only one that is both correct and common is #4.
In Case 2, the ones that are both correct and common are #1, #4, and #5.
However, you would never use #5 unless you knew that Mr. Smith likes to be referred to in this way. Some people adopt their middle name as their "handle", and downplay their given first name. Most do not.