I read somewhere that when chessplayers get together and start naming world champions, the name most often left off the list is Vasily Smyslov! Guess what---YOU LEFT HIM OFF YOUR LIST!
LIst of world champions


And I don't agree with the end of your list : Kasparov didn't loose in 1993, but beat the winner of candidate matches Nigel Short.
So your list should be :
FIDE world champion :
15.Anatoly Karpov - Russia 1993-1999
16.Alexander Khalifman - Russia - 1999-2000
17.Viswanathan Anand - India - 2000-2002
18.Ruslan Ponomariov - Ukraine - 2002-2004
19.Rustam Kasimdzhanov - Uzbekistan - 2004-2005
20.Veselin Topalov - Bulgaria - 2005-2006
Classic world chapion :
14. Garry Kasparov - Russia - 1985-2000
15. Vladimir Kramnik - Russia - 2000-2007
16. Viswanathan Anand - India - current world champion

WILHELM STEINITZ (born May-17-1836, died Aug-12-1900) Austria (citizen of United States of America) |
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Wilhelm Steinitz was the first official World Champion of chess.
Background The last of thirteen sons of a hardware retailer, he was born in Prague in what was then the Kingdom of Bohemia within the Austrian Empire and which is now within the Czech republic. Like his father he was a Talmudic scholar, but then he left to study mathematics in the Vienna Polytechnic. He eventually dropped out of the Polytechnic to play chess professionally. Soon after he played in the London tournament of 1862, settling in London for over twenty years and making his living at the London Chess Club. He emigrated to the USA in 1883, taking out US citizenship, living in New York for the rest of his life, and changing his first name to “William”. |

I don't really think Americans can claim him but he did become an American citizen in the last years of his life . I consider him Austrian myself.

FIDE world champions 1948–1993[change]
Long matches, initially best of 24 games.
# | Name | Year | Country |
---|---|---|---|
6 | Mikhail Botvinnik | 1948–1957 1958–1960 1961–1963 |
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7 | Vasily Smyslov | 1957–1958 | ![]() |
8 | Mikhail Tal | 1960–1961 | ![]() |
9 | Tigran Petrosian | 1963–1969 | ![]() |
10 | Boris Spassky | 1969–1972 | ![]() |
11 | Robert J. Fischer | 1972–1975 | ![]() |
12 | Anatoly Karpov | 1975–1985 | ![]() |
13 | Garry Kasparov | 1985–1993 | ![]() ![]() |
1.Wilhelm Steinitz - USA - 1886 - 1894
2.Emanuel Lasker- Germany - 1894-1921
3.Jose Raul Capablanca- Cuba - 1921 - 1927
4.Alexander Alekhine - France - 1927-1935
5.Max Euwe - Dutch(netherlands?) - 1935-1937
6.Alrxander Alekhine - France - 1937 - 1946
7.Mikhail Botvinnik - Russia - 1948-1957
8.Mikhail Tal - Latvia - 1960-1961
9.Mikhail Botnivik - Russia - 1961-1963
10.Tigran Petrosian - Armenia - 1963-1969
11.Boris Spassky - Russia - 1969-1972
12.Robert J Fischer - USA - 1972-1975
13.Anatoly Karpov - Russia - 1975-1985
14.Garry Kasparov - Russia - 1985-1993
15.Anatoly Karpov - Russia 1993-1999
16.Alexander Khalifman - Russia - 1999-2000
17.Viswanathan Anand - India - 2000-2002
18.Ruslan Ponomariov - Ukraine - 2002-2004
19.Rustam Kasimdzhanov - Uzbekistan - 2004-2005
20.Veselin Topalov - Bulgaria - 2005-2006
21.Vladimir Kramnik - Russia - 2006-2007
22.Viswanathan Anand - India - 2007 -2008
Female World Championship
1.Vera Menchik - Czehslovakia - 1927-1944
2.Lyudmila Rudenko - Soviet Union - 1950-1953
3.Elisabeth Bykova - Soviet union -1953-1956
4.Olga Rubtsova - Soviet union - 1956-1958
5.Nona Gaprindashvili - Soviet Union-1962-1978
6.Maya Chiburdanidze - Soviet Union-1978-1991
7.Xie Jun - China - 1991-1996
8.Susan Polgar -Hungary/USA - 1996-1999
9Xie Jun - China - 1999-2001
10.Zhu Chen - China - 2001-2004
11.Antoaneta Stefanova - Bulgaria - 2004-2006
12.Xu Yuhua - China - 2006-2008
13.Alexendra Kostenjuk - Russia - 2008
Got the list off a local book