The newspaper Savannah Daily Morning News, which William Tappan Thompson edited, published an article Confederate Flag and Seal on 23 April 1863.
The article advocated for a particular flag design and contains the first OP quote. A fuller version of the first OP quote makes clear what the supremacist symbol is:
As a people, we are fighting to maintain the Heaven ordained supremacy of the white man over the inferior or colored race ; a white flag would thus be emblematical of our cause
Also, the article uses the first person plural, taking at least partial credit for the design:
Our idea is simply to combine the present battle flag with a pure white sheet
However, at the end of the article the following note is added:
Since the above was in type ... the senate has adopted a new flag... with the exception of a blue stripe in the centre of the white field, the new flag is the same as we have suggested above
The book Our Flag (1872) further explains that in the House, Julian Hartridge introduced an amendment which was passed changing the flag (relative to the Senate bill) to have an all white background, without the blue stripe.
(Julian Hartridge practiced law in Savannah and represented Georgia in the House)
With foreknowledge of Julian Hartridge's plan, but before the amendment was introduced by Hartridge and with expectation of the House and Senate having approved different flags, Savanna Daily Morning News published another editorial The New Confederate Flag 28 April 1863. This article further expressed the supremacist symbolism of the all-white background flag saying (with all caps in the original text as below):
hailed by the civilized world as THE WHITE MAN'S FLAG
In the meanwhile (April 24) P. G. T. Beauregard also advocated the same design in a letter, but without expressing any supremacist reasons, and in fact writing "all white or all blue".
The Journal of the Confederate Congress, vol. 6, page 476 confirms that on 1 May Julian Hartridge did move to amend the flag bill to have the all-white background, fulfilling Thompson's prediction published four days earlier. Other amendments by others were defeated and the all-white background flag was approved that day.
The 1872 Our Flag book specifically credits Thompson, saying:
His remonstrance against the addition of the blue bar appear to have been heeded.
The second OP quote is from a 4 May 1863 article reporting the approval of the flag.
A couple weeks later, a 20 May 1863 Savanna Daily Morning News article The Flag stated:
As a symbol of the white man's cause, long may it float in triumph
In conclusion, Thompson had no role in the design of the Battle Flag which is shown in the OP graphic, but he wanted to make the CSA national flag be the Battle Flag on a white field, for white supremacist reasons, and the CSA congress did make this the official national flag.
For very extensive information with numerous color drawings of proposed flags and hundreds of newspaper excerpts see ILLUSTRATED DOCUMENTARY HISTORY OF THE FLAG AND SEAL OF THE CONFEDERATE STATES OF AMERICA. 1861 - 65 (1880). Numerous flag designs were considered before the flag as introduced by Hartridge was adopted. Some people gave extremely racist reasoning for their designs, with white representing the supremacy of the white race, while to others white represented, purity, innocence or peace.
(A comment says there is confusion about what flag is being discussed, so I'm adding an example of the flag:)
(historical flag located at the George Washington Masonic Memorial, Alexandria, Virginia, as of 2011, image source )