To some degree, the logo can help
There are 5 general logos or 8 if you include GK and logo variants that allow us to identify especially the early times of TSR. This can help to date a print into rough eras.
Sometimes more has to be considered
A more detailed analysis on any printed product needs to include the coloration and picture chosen, sometimes together with other markings. For this, the trade-value catalog acaeum.com for pre-1990 TSR material can help very much to find the specific markers that differentiate different print runs on products.
Deities & Demigods is special
This leads for example to the realization, that the Deities & Demigods changed its cover on the official 3rd printing in 1981, helping to date the product variant in the OP - it indeed was always meant to be blue-shifted compared to the first printings.
Of special note about that manual is, that what went to sales as the official "first print" edition technically consists of two print runs, one that was halted midway for lacking a specific permission note (treated as 1st print by collectors) and one that had it (treated as 2nd print by collectors) that was to be the real edition sold first. The two together make up about 10-15 000 prints remaining today. They were not considered separate for the printing of the 15-pantheons-included-17-on-back version in about 1980/81 (which was considered 2nd printing by the Blume Brothers) but would be counted separately for establishing the December 1987 print as the sixth in the internal cover.
Also note that for this specific book, it is unclear if the print in question belongs to a 4th or 4+ printing (using the collector parlance) due to the most notable difference between the two being the removal of a credit line on page 4. It is unclear if those are distinct print runs shortly after one another or a mid-print alteration.
The fifth run (1984) onwards uses a different title, cover art, and logo, thus dating the found example to about 1981 to 1983.
GK
Before the TSR Logo, there was the Gygax-Kaye one. That of course stands for Gary Gyax and Don Kaye, who formed a partnership of authors and publishers. The logo is two wrought-iron-styled letters, G and K, intertwined, usually joined by the full title of the partnership: Tactical Studies Rules. It was officially used from 1974 to 1975, such as on Chainmail 3rd Edition, 1st Print, which came out before July 1975 (when the 2nd print ran). The same logo setup can be found on the first Dungeons and Dragons products, but labeled with a different price.
The abandonment of the logo came soon after the death of Don Kaye. However, some products might have still been printed or at least sold with the GK logo till about 1979. Gary Gygax and Brian Blume reorganized the partnership into TSR Hobbies Inc. and bought out all other partners, such as Don Kaye's widow and Dave Arneson.
The Lizardman + TSR
The first logo of TSR Hobbies Inc. (originally owned by Gary Gygax and Brian Blume and partially funded by Brian's father Melvin Blume) was a lizardman holding a Helbard, and the shorthand TSR. It was used between July 1975 and December 1978. One example of use is the Chainmail 3rd Edition, 3rd Print of June 1976 as seen below, next to a more clear rendition of the lizardman.
Wizard Logo
Starting December 1978 to Mid/Late 1980, the Logo of a wizard in robes with a pointy hat next to the text "TSR The Game Wizards". The font is serifless. This logo can be observed on the first print of Deities & Demigods.
Face & TSR cartouche
In or about late 1980, the style swapped to a face-coin and the letters TSR in a Cartouche using a slightly cursive font with a subscript "The Game Wizards" using the same font.
This indicates that the right picture in the OP is not from the initial publication runs. As elaborated above, it's most likely from the third official (technically fourth) print run, but it was also made before mid/late 1982, when the logo was discontinued.
TSR stairs/slope
After the face cartouche, from late 1982 to 1991, TSR used the letters of its name arranged akin to steps in downward stairs, or sloped downwards. This is the longest and most iconic logo rendition. Note that the logo survived the 1986 sale to new owners.
TSR stairs/slope in gold cartouche
From 1991 to 1992, TSR put its logo into a gold cartouche or plate. This was short-lived. For example, the 1992 catalog featured this logo (see below). It can be considered a variant of the TSR stairs/slope.
TSR dragon coin
For products from late 1992 to 1999, a totally new logo came around: A red dragon growing from the silver letters TSR on a gold-framed coin. Technically the company became an asset of Wizards of the Coast in 1997, but some product was branded TSR under this management.
Anniversary TSR dragon coin
For the silver anniversary coming, TSR altered its logo in 1999 to a silver coin with a wider rim reading Silver Anniversary. It stayed till 2003 as a brand of (then) Wizards of the Coast. It can be considered a variant of the TSR Dragon Coin.
Outliers
It is of note that logos were not always consistent with the most current logo of new products when it came to late reprints.
The 7th print of S1 Tomb of Horrors was sold in 1999 with the Wizard Logo, after having had 3 prints with the Face Cartouche. It was an anniversary print though and had been out of print for more than a decade before then.
Similarly, such late prints can show up with multiple logos. For example, the 10th Print of S2 White Plume Mountain shows both the Anniversary Dragon Coin and the Wizard Logo. We know that this print is from 1999.
