No. Two of the first three movie producers to move out to L.A. were Selig Polyscope and Biograph, both of whom were members of the Edison Trust.

Three more Trust member producers--Vitagraph, Essanay, and Kalem--had all started production in the L.A. area by 1911.

Independent producers, in fact, didn't move to L.A. to get away from the Edison Trust. They followed them out there!

The guy who played the biggest role was the head of Selig Polyscope, William Selig. His production company was based in Chicago, but he sent a film crew out to Colorado a few months every year to make Westerns. Selig was a firm believer in using natural locations in order for his films to look more believable. He was on to something, because Polyscope was the second most profitable film producer in the U.S. before 1910, behind only Vitagraph.

These Westerns were so successful that, in 1906, Selig started exploring the possibility of setting up a permanent West Coast operation. This led to him sending out a film crew to L.A. in March 1909. Polyscope rented out the backyard of Sing Kee's Chinese laundry for three months, located at 751 S. Olive Street in downtown Los Angeles.

The film crew reported back that filming conditions were ideal, and the available talent and filmmaking materials were passable. So when the Sing Kee lease was up, the film crew traveled up the Pacific Coast making more Westerns, while William Selig had a representative find a suitable lot in L.A. to purchase.

Located at 1845 Allesandro Street, Selig Polyscope bought the first movie lot in Los Angeles history on August 26, 1909. By the end of the year, Polyscope had produced fifty one-reel shorts in Los Angeles. They issued a press release indicating that Polyscope would now permanently be producing two one-reel films every single week--one at their Chicago studio, and one at their Los Angeles studio. By the end of the next year, Polyscope had finished building a permanent indoor studio on their L.A. property to go with their outdoor lot.

Shortly after the initial purchase, in September 1909, William Selig was overseas in London meeting with European film distributors interested in his library of films. While there, he gave an interview with the Kinematograph & Lantern Weekly which was reprinted in New York's The Film Index that same month. In the interview, Selig explained that the move to Los Angeles was for the realism the natural landscape and scenery of the area offered:

"Our studios and stables in Chicago extend over five acres, and we have another five acres in Los Angeles, where we get our best scenic effects from. We believe in giving the public the most realistic picture we can get, and we're spending a lot of money on it, too. But it will come back."

Selig was followed out there almost immediately by other companies. By the end of the year, an upstart independent company called the New York Motion Picture Company bought a lot right down the street from Polyscope, which was weird since it was on a dirt road three miles outside of downtown L.A.

Why would a non-Trust movie producer move so close to one of the Trust members? MGM director Robert Leonard later offered up an explanation. He had been an actor for Polyscope that came out with the first crew in 1909. In a 1939 Chicago Tribune interview, he explained that in those days, when he wasn't acting, he was tasked with "patrolling the film location armed with a loaded gun to 'scare, if not shoot' cameramen from independent companies attempting to surreptitiously film the Selig production and pass it off as their own."

Bison was the imprint that the New York Motion Picture Company used for their Westerns. Funny they should open up right down the street from Polyscope, the premiere Western producer in the world at that time (before the term "Western" had come into general use).

Biograph, another member of the Edison Trust, arrived in L.A. in January 1910. They came partially for the same reason--to shoot Westerns in cooperative weather--but that wasn't the only reason. By 1910, the big Eastern film companies were big enough that they needed ample studio space in order to have multiple film crews operating year-round. Real estate was expensive in New York and Chicago. But in L.A., even though real estate was booming, a whole ranch or farm could still be bought at an excusable price.

Thus, the Moving Picture World reported in their January 29, 1910, issue, about Biograph's move:

"Having exhausted all the facilities of their cramped but well utilized studio on Fourteenth Street [in New York City] they realized that they were handicapped in futher progress, therefore another site was looked for. After careful study of the advantages of several locations a plot of land in the Bronx was selected as the ideal spot...

"But this is not all. Acute minds perceived that the march of progress must not await the new habitation; the rural settings around [metro New York] had been well nigh exhausted: the inclemency of the season made journeys into the country a severe task. So on Wednesday, January 19, [1910,] the entire Biograph stock company, with [D.W.] Griffith, director-in-chief, was dispatched on fast trains to the sunny and picturesque lands of Southern California. With headquarters in Los Angeles, where a developing plant has already been installed, with a careful selection of scenarios suitable for that region and with the inspiring influence which a delightful climate must exert on the artistic temperament, we may look forward to seeing on the screen, pictures that will yet more and more maintain the Biograph name for quality."

Other East Coast companies, who had done seasonal shooting in previous years in places like Florida, quickly saw the advantages. Movies were already big business, so any move out West was met by small town governments in Southern California clamoring to give these businesses favorable tax and real estate deals.

In 1914, Carl Laemmle, the head of Universal Pictures, took an ad out in the L.A. Times, offering to move his studio to whichever Southern California town gave him the most favorable financial terms. This led to Laemmle buying more than six hundred acres of land on the edge of L.A., which he renamed "Universal City" when it opened later that year. If Los Angeles wasn't already synonymous with the movie industry before that, it was then.

The term "Hollywood" and the Hollywood section of L.A. specifically being synonymous with the movie industry came a little bit later, in the 1920s.

Over the years, many legends have grown up around the start of the film industry in L.A., with several different people claiming responsibility. These include the aforementioned D.W. Griffith as well as Cecille B. De Mille. Misinformation surrounding the Edison Trust's role in the move to L.A. has circulated since the 1930s. A lot of the above information comes from the meticulously researched book Col. William N. Selig: The Man Who Invented Hollywood by Andrew A. Erish. The author gives a rundown of how this Ediston Trust misinformation got its start:

In his 1931 History of the Movies, Benjamin B. Hampton recognized Selig for having "discovered Los Angeles," but incorrectly identified him as an independent filmmaker whose motivation was to "produce pictures in places so remote that [Edison] subpoena-servers and confiscators of cameras would have trouble finding his troupes." Hampton claimed that other independents "investigated Selig's operations, and unhesitatingly settled in and near Los Angeles because it was close to the Mexican border." The widely regarded 1938 History of Motion Pictures, written by Frenchmen Maurice Bardeche and Robert Brasillach, elaborated on Hampton, asserting that "if [MPPC] detectives turned up, [the independents] could pile actors, scenery and cameras into a car and disappear across the border for a few days." Both Kenneth Macgowan's 1965 History and Jeanne Thomas Allen's chapter in Tino Balio's influential American Film Industry, published in 1976, repeat earlier claims about Selig as a former independent leading other outlaw filmmakers to Los Angeles, while admitting ambivalence about the obvious contradictions. The claim about Mexico's proximity to Los Angeles is especially ludicrous; L.A. is approximately 150 miles from the border, and in 1909 was reachable only by railroad or dirt roads traveled by horse and wagon, negating any fast getaways, not that there was ever any need for them...

"In the parlance of 1911, William Selig had become a 'moving picture magnate.' The Selig Polyscope Company was second only to Vitagraph as the most profitable American motion picture firm. Because Selig went to Los Angeles, others followed."

TL;DR: L.A. became the film industry capital of the United States due to scenery, sunshine, and real estate. It was also easy to get to, being the terminus of the Southern Pacific Railroad. The first movie producer to move there was a member of the Edison Trust, and four more Edison Trust members had arrived within the next two years. Independent producers didn't go there to escape the Trust. They followed the Trust member producers out there. At the time this relocation was occurring between 1909-1914, Edison's harassment of smaller producers had largely been curtailed by court order, which is why producers like Universal and Paramount were able to get their start even before the Supreme Court dissolved the Edison Trust in 1915.