Correct Variant ID

I am in a number of forums, but for some reason this is the first one in which this question seems appropiate. I am familiar with Willy (pronounce Villy ) Messerschmidt and a lot of his aircraft, however; which in correct is it the Me-109 or Bf-109? ...I know most folks call it the Me109 probably because it's build by Villy and the kids...,and that's a fact, but how come or where did the Bf come in and why......an inquiring mind vants to know Ja.
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The "Bf" designated the factory which first manufactured it "Bayerische Flugzeugwerke", however I believe either one is correct.

Bill Gunston wrote a foreword for an anniversary edition of Jane's 45-46 and in it he chastises Bridgeman for interposing Me and Bf...

With VB... the Bayerische Flugzweugwerke (Bavarian Aircraft Works) designation Bf was used originally, Willy Messerschmitt's preference if I remember correctly, later redesignated Me by the RLM (in 1941?) to bring it in line with their standardised pattern of manufacturer designation (Dornier -Do, Blohm und Voss - Bv, etc). Foche-Achgelis had theirs changed from Ha to Fa aswell, for the same reason I believe.

The Me 155 is the first Messerschmitt design registered as such by the RLM, and all types from this RLM register number on carry the designation (Me 155, 209, 210, 309, 321, 323, 328, 410...), with the exception of the Bf 161 and 162, these numbers being specially requested by Messerschmitt prior to the designation change-over. (The RLM reg. no 162 was later awarded to Heinkel after the failure of the Bf 161 project).
Modified versions of existing types seem to have carried (atleast unofficially) the Bf designation to the end - Bf 108, 109, 110...although to the RLM's mind atleast these should be 'Me's from the 109G, and K series, and 110E or G series on...

(This is all as far as I've read from various sources over the years)