So there is an Era, TX but it was 150 years away in 1710. So lets look at Cooke County that it sits inside, which originally encompassed the surrounding counties as well. It was 140 years later, but like you said it was Spanish land back then. However according to historians, aside from the random wandering Spanish expedition, no Europeans were in the area and no settlements occured until the 19th century. It is very highly unlikely that your folks were typical English settlers on a Spanish expedition.

According to the Texas State Historical Association;

Before the coming of Anglo-American settlement Cooke County stood on the borderlands between the Caddo Indians to the east and the Comanches in the west. The first Europeans to visit the county may have been Spaniards on expeditions during the sixteenth, seventeenth, and eighteenth centuries, but no permanent settlements were made. The county was included in the Cameron land grant, a Mexican grant of 1828, but no settlers came.

Let's keep looking.

Spanish missions... Didn't go that far north (ever). They got a few hundred miles away by 1795, so that doesn't help much to explain anything. Additionally, Spanish records indicate native raids from North and East Texas which would include what became Era.

French were in Louisiana. The Spanish got real mad and threatened them but didn't take action. They became partners just before then during Queen Anne's War which continued until after 1710, so an English family going to settle enemy land in the middle of a war would be a bad idea. Further, even though they were friends, the French had moved into current Texas to trade a few years before the birth and in response the Spanish actively tried to stop them coming even just to trade. After America (British) gained land to the Mississippi (1763) they still resisted settlement but eventually permitted it in "French Louisiana" (which they controlled for a while) and Florida (after getting it back from the English) but not Texas. So we see active resistance to any non-spanish settlements in Texas until the 19th century as well. Some English colonists went there anyway, to "steal" wild horses for instance, but those records don't appear until the late 18th century and aren't in that area of Texas. It's noteable that the Spanish wouldn't have known your fam was there as it was native controlled and not spanish controlled.

Ok the Spanish and French don't look like good clues, let's see what British North America was doing in 1710. Trading was happening in Kentucky but permanent Anglo settlements there were still 50+ years away. Tennessee was a few years from having a fort built that eventually became Nashville. South Carolina was occupied with their own issues and James Oglethorpe hadn't yet conceived of the financial undertaking he planned and executed called Georgia (1733). Ben Franklin was 3 in Puritan Boston. George Washington's mom was 2. The founders weren't yet born. The Salem Witch Trials had ended not very much earlier.

It seems extremely unlikely a pregnant Anglo settler was in North central Texas in 1710. Even more unlikely is staying there a decade. The biggest question I have, however, is how this record came to be. If he was born in 1710, WHO wrote down Era and where did they get the information to connect a birth in a spot to the name attached to said spot 150 years later?

What would an Anglo settler be doing in that part of Texas in 1710? Scratching his head about where the hell he is and how he got so lost while being very concerned about native warriors and tribes.

What is that source claiming the birth? Happy to help privately if you would rather message me.