We have the connection "Once the tallest freestanding structure in the world". Such lists often distinguish structures from buildings. Buildings are structures that can be regularly inhabited or occupied, so for example the Giza Pyramid is excluded.
In our connection we specifically opted for structure. This means that the Giza Pyramid is included (146.5 m). This would mean that Straßburg Cathedral (142 m) and St. Olaf's Church in Tallinn (125 m) and should be out.

It would be helpful to have the height added to the connection:

Giza Pyramid 146.5
St. Mary's Church in Stralsund 151
Cologne Cathedral 157.5
Eiffel Tower 324 m

I think it would be interesting to add a different connection "Once the tallest building in the world". This would exclude the pyramids as well as Eiffel Tower and would hence result in quite a different list of monuments:

Etemenanki (Babylon) - 601 BC-323 BC (91 m)
Pantheon (Rome) - 126-516 (43.5 m)
Haghia Sophia (Istanbul) - 537-late 7th century (55 m)
Hwangnyongsa (Gyeongju) - late 7th century-823/1238 (68 or 80 m)
St. Olaf Church (Tallinn) - 1519-1625 (159 m)
St. Mary Church (Stralsund) - 1549-1569, 1573-1647 (151 m) (If Tallinn is accepted then only 1625-1647)
Straßburg Cathedral - 1647-1874 (142 m)
Cologne Cathedral - 1880-1889 (157.5 m)

Assif
I think the structure vs building definition is quite tricky and totally artificial.
For example, in my opinion, the pyramid of the sun (in teotihuacan) had a quite similar purpose and usage than etemenanki, and is much higher than the pantheon or the hagia sophia (from wikipedia: height 65.5m, building year ~220). In contrast to the egyptian pyramids, its main use was religious worship, just like a church.

Also, the years between 1238 and 1519 are not really covered in your list. Possible contenders are the Minaret of Kutubiyya Mosque, Marrakesch (77m from 1195-) and Lincoln cathedral (159m from 1311-1549, but not a WHS ?).

I also totally fail to see why the eiffel tower should not count as a building. It holds plenty of commercial space, shops, restaurants, even a small office at the top, has a completely commercial, non-technical purpose, is accessible by the public and has a continuous high occupancy rate. You can't sleep in it, but that's true for an office tower, too.

Things are certainly easier when we stick to the "tallest structure" definition...

I'm convinced. The distinction is indeed difficult. I thought it has to do with the usage of inner rooms, which are sometimes almost entirely missing, for example in many pyramids. Nonetheless, a structure is easier to define (although it still has to be distinguished from a mere structure).
If we stick to the existing connection Straßburg should be out as it is smaller than the Giza Pyramid. As to St. Olaf, Tallinn, I see there are different sources with varying estimates of its height in 1519-1625 ranging between 159 m (which would entitle it to be included in this connection) and 125 m (which would not). Any expert opinions?

From Finnish Wikipedia (Olevisten kirkko):
"Originally raised to a height of 159 meters, the tower was probably the second tallest of the European buildings of its time. Lightning ignited a fire in 1625, after which the length of the Church tower was shortened by twenty meters." ... "But because of the fire, its current height is 123.7 meters."

@Sjobe: the English Wikipedia provides different data:
"In 1590, the total height of the church tower was 115.35–125 m. The tower has been hit by lightning around 10 times, and the whole church has burned down three times throughout its known existence. According to a source it may have been the tallest building in the world from 1549 to 1625, but this claim is disputed."
If the disputed height of 159 metres is true, then the church was the tallest building in world at the time (surpassing St. Mary of Wismar), not only the second tallest in Europe.

Well, I didn't know the height of St. Olaf, Tallinn is so controversial as it seems to be.

From Estonian Wikipedia (Oleviste kirik), translation by Google Translate:
"Together with a Gothic-style tower helmet completed around 1500, the church was 115-125 meters high." ... "It is said that the Oleviste Church was the tallest building in the world in 1549–1625. This opinion is based on a letter found in the church tower in 1778, according to which the height of the church tower was 84 fathoms before the fire of 1625. However, it is not clear what kind of brood is intended. This corresponds to 159 meters in the Rhine, 161 in the Hamburg, 179 in the Russian, and 134 meters in the Tallinn. No other source proves this number, and according to the available images, the tower of Oleviste Church has always been about as high as it is today. Also, in the case of a high tower, the stone part would be disproportionately low." ... "On the night of May 29, 1625, an elongated church bell tower ignited. The tower, church bells and all the furniture were destroyed. Only the walls survived. The church was quickly restored, and the doors were opened for worship three years later. The new bell tower was completed in 1651, it was built 135 meters high. Also in 1693, 1698, 1700, 1707, 1719 and 1736, damage to the tower and minor fires as a result of lightning strikes are noted."

And here is an article on an Estonian website: Was Oleviste Church ever the tallest building in the world?

I've updated the connection with the info above. Thanks Assif, Ralf and Sjobe!

Based on these sources, we could quite safely say that the height of Oleviste Church has never been 159 meters, and thus it could be wiped out from this connection.