You pretty much have it down pat.

The reason you are confused is you are conflating two different meanings of the world “Celtic.”

Nowadays, Celtic means what you say in the post. It has a common meaning associated with Scottish and Irish culture. However, the anthropological/historical definition is much different (although related.)

Going back to the Iron Age, “Celts,” a large group of Indo-European people, had migrated across Europe in swathes, setting all across Britain, France, Northern Italy, part of Spain, into the Balkans, and western Turkey (Anatolia). These were not a monolithic group, but highly decentralized tribes with no greater sense of unity as a collective.

When IVLIUS CAESAR invaded Gaul, he was conquering Celts. Gauls were Celts, and Gauls also were highly disparate and lived in small tribal communities. There were hundreds of these tribes like the infamous Arverni, the Parisii around Lutetia (now called Paris), and hundreds more.

It may be useful to think of it with his analogy: French people and German people are Europeans. People from Paris, Bordeaux, Marseille, and Rouen are all French.

Despite sharing these huge labels, significant cultural differences exists on every level of these divisions.

Lastly, the reason the word Celt is associated with Scottish, Irish, and Breton culture is because they are the last surviving groups that still speak Celtic languages. They are the last survivors bottled up in geographically extremes on the fringes of Europe.

So, in short, the Celts of the ancient world, of which the Gauls were part of, have very little in common with those of modern day. This is because we cannot consider the word Celt as an unmoving, monolithic term.

Now, a little more about Gauls. Gauls were adept swordsmen and great at ironworking. The Gauls around Marseille had contact with Greeks (Massilia was a Greek colony), and learned fine arts like how to use marble from them. When Rome conquered Gaul, the Gauls became Romanized, mixed Latin with their native language, and inherited the fruits of Roman civilization. Thereafter, Gauls became Gallo-Romans, a term reflecting the cultural mixing. The earliest version of French is Old Gallo-Romance, which you can read about by researching The Sequence of Saint Eulalia and The Oaths of Strasbourg (year 842).

When the Franks defeated the rump state of Soissons and took over Gaul under King Clovis, they, as with other barbarian groups who invaded Roman territories, were conquered in turn by Roman legacy and culture. The Franks settled into Gallo-Roman society, and became appropriated. Of course, there are a few thousand words in French of Frankish origin, but the Frankish language was absorbed into the Gallo-Roman one.

Charlemagne set his clergy and educated men to the task of revitalizing education and latin texts, and created the Carolingian Renaissance. This is also a key moment for French heritage because it further promoted Latin heritage and customs. Also, the Franks adopted and mixed existing Gallo-Roman cultural practices and language into their own, creating Old French.

In short, Celtic Gauls became Gallo-Romans, who, in turn, Gallicized and Romanized the Franks who, in turn, contributed to the region creating the Old French Language and a new culture that has grown into what is now French.