словосочетание ваш искренне - это всего лишь достаточно частая форма вежливой
самоадресации 
в англоязычной (особенно британскоанглоязычной) переписке... в данном случае ваш искренне = yours truly
Well, there are different schools of thought nowadays. However the basic rule is that when you use Dear Sir/Madam then you end with yours faithfully. When you know the name of recipient then use yours sincerely.
General example:
Dear Mr Brown,
blah blah blah.
Yours sincerely,
Me.
When addressing a close friend or colleague and you use the first name, then it's:
Dear Jessica,
blah blah,
best wishes, warm wishes, best regards, warm regards, kind regards,
Me
Increasingly, especially in the United States, people are signing off with best / kind / warm regards regardless of the form of address.
Although all the above is correct it can become somewhat of a minefield when entering the realms of Perigee, Royalty & other titles; For example, if writing to your mayor it should set out as follows if you live in a City:
The Rt. Worshipful The Mayor of
(Councillor )
The Mayors Parlor
City Hall
Dear Mr/Madame Mayor, (Change as required
Respectfully Yours,
Joe Bloggs
HOWEVER this is not true if you live in a Town or Borough:
The Worshipful The Mayor of
(Councillor )
The Mayors Parlor
Town Hall
Dear Mr/Madame Mayor, (Change as required)
Respectfully Yours,
Joe Bloggs
Note that if in a City the Mayor is Worshipful in his/her own rights (The Rt. Worshipful), otherwise the Mayor is appointed as worshipful (The Worshipful). Also note that the Mayors forename is NEVER used (extremely rude to call the Mayor by his/her first name in public) and even if you know the Mayor you always end the letter with Respectfully yours (irrespective of gender or location)
(c)
Yours truly -- книжное, весьма устаревшее в самой Британии.
. ( 
