It's not partly true at all, it's completely false. The 1889 Chigorin-Steinitz match took place in January and the New York tournament took place in March.

<He declined to challenge at that time, having just lost his first match to Steinitz the year before. Weiss declined too, so Gunsberg got it. Then, 3 years later, Tchigorin challenged again.>

Total nonsense! Steinitz had resigned the title during the run up to the 1889 New York tournament which officially stated in it's regulations that it was a World Championship tournament, and that the winner would be World Champion. The second place finisher had the right to challenge the winner to a title match if he so chose. This had the full support of Steinitz who was one of the chief organizers and promoters of the tournament. Steinitz had this to say on the subject:

"I know I am not fit to be champion and I am not likely to bear that title forever. Therefore let us select a better one, and whoever it may be, I shall cheerfully say to him, like the old soldier: My son, here you have my spear, it becomes too heavy for my arm."

"Anyhow, I may positively state that if I do not take part in the main contest I shall not challenge the winner...the ultimate winner, provided that he fulfills all the conditions of the Committee shall have my most loyal support for his Champion title to which I shall lay no claim..."

It is clear that the New York tournament of 1889 was not meant to select a challenger to Steinitz, but to determine the new World Champion. Steinitz himself was one of the chief movers behind this tournament and was 100% behind it's World Championship status.

After the tournament Weiss went into virtual retirement from chess and returned to his banking career. Gunsberg (3rd place) did indeed challenge Chigorin (1st place) in 1890 to a match with identical terms to Steinitz-Zukertort 1886. The match took place in Havana and ended in a 9-9 tie (ala Fischer). According to the New York Tribune the Chigorin-Gunsberg match was for “the championship of the world”.

The Steinitz-Gunsberg match however, was an informal match (limited number of games with draws counting toward the final score) and not a World Championship match (unlimited number of games and draws do not count). Nowhere in the match contract is the world championship mentioned. In fact here is the exact wording from the contract:

I'm glad to see you agree that New York 1889 had absolutely nothing to do with Chigorin-Steinitz 1889. Where you are mistaken though is in calling the Chigorin-Steinitz matches "championship matches." Both 1889 and 1892 were 20 game matches with draws counting toward the final result (1892 had a 3 game tie break at the end). Steinitz explicitly stated that such matches were not fit for a world championship contest and he seems to have employed this format when playing informal matches during the 80's and '90's.

From <Edward Winter's Chess Notes>:

<Little-known nineteenth-century occurrences of the term ‘grandmaster’ are always welcome. Robert John McCrary (Columbia, SC, USA) forwards one from page 324 of the <Chess Player’s Chronicle, 1854>:

And don't even mention: "In 1914, Tsar Nicholas conferred the titles of Grandmaster on the top five finishers at the St. Petersburg tournament."

The match took place in the Hotel Metropole in Brighton, from August 3 to 15, 1903. Dr. Emanuel Lasker had the White pieces in every game as they had to play the Rice gambit. Chigorin won +2 =3 -1 as is reported on page 224 of the 1903 'Wiener Schachzeitung'.

This was not a competitive match as Chigorin also emphasized in the 'Novoye Vremya' but a research on the Rice gambit. They played 4 sessions a week, with 1 hour for 10 moves and not more than four hours per session (page 250 of the 1903 'Wiener Schachzeitung').

Here are the games (with dates accoring to pages 251 to 252 of the 1903 'Wiener Schachzeitung' on the game pages):

Game 1: Lasker vs Chigorin, 1903

Game 2: Lasker vs Chigorin, 1903

Game 3: Lasker vs Chigorin, 1903 (ended after 29...Kg7)

Game 4: Lasker vs Chigorin, 1903

Game 5: Lasker vs Chigorin, 1903