I also have wondered what that phrase meant exactly...'they began to call upon the name of the Lord' and posed it in another forum some years back. Several people responded and every one of them understood it to have a negative connotation. Checking other sources, they seem to agree with that although as you say, it isn't 100%.
I get tired of the manipulations of scripture people use to 'prove' what they think it should say when the script reads pretty plain to begin with.
I am not sure which you are thinking are trying to manipulate, those who say it means that they begin to call upon the name of the Lord, or those that say it means profane the name of the Lord.
All the best Biblical Textual Experts and Hebrew Scholars (entire committees of scholars who are behind these English translations) The best available in the world, have not concluded that it has negative connotations.
If there was sufficient evidence that it should be negative it would have been decided as such by all of these committees. I have a strong feeling that when I take three years of Hebrew and by then will be able to barely read the original manuscripts for myself I will probably agree with all the following committees behind these translations:
Only one out of these 27 translations thought that profane should be used.
Do I really think that is the correct one over the other 26 (and this is not the full list)? No I do not. I rely on experts for now. Later maybe I can explain why they were right but I doubt I will discover at that time that they were wrong. Very highly unlikely. Don't you think?
I mean if I were to pick the one out of the 27 to prove that it meant (profane the name of the Lord) that would reek of dishonest manipulation to find one that agrees with what I want it to say. I agree with that.
New International Version
Seth also had a son, and he named him Enosh. At that time people began to call on the name of the LORD.
New Living Translation
When Seth grew up, he had a son and named him Enosh. At that time people first began to worship the LORD by name.
English Standard Version
To Seth also a son was born, and he called his name Enosh. At that time people began to call upon the name of the LORD.
Berean Study Bible
And to Seth also a son was born, and he called him Enosh. At that time men began to call upon the name of the LORD.
King James Bible
And to Seth, to him also there was born a son; and he called his name Enos: then began men to call upon the name of the LORD.
New King James Version
And as for Seth, to him also a son was born; and he named him Enosh. Then
men began to call on the name of the LORD.
New American Standard Bible
To Seth also a son was born; and he named him Enosh. Then
people began to call upon the name of the LORD.
NASB 1995
To Seth, to him also a son was born; and he called his name Enosh. Then men began to call upon the name of the LORD.
NASB 1977
And to Seth, to him also a son was born; and he called his name Enosh. Then
men began to call upon the name of the LORD.
Amplified Bible
To Seth, also, a son was born, whom he named Enosh (mortal man, mankind). At that [same] time men began to call on the name of the LORD [in worship through prayer, praise, and thanksgiving].
Christian Standard Bible
A son was born to Seth also, and he named him Enosh. At that time people began to call on the name of the LORD.
Holman Christian Standard Bible
A son was born to Seth also, and he named him Enosh. At that time people began to call on the name of Yahweh.
American Standard Version
And to Seth, to him also there was born a son; and he called his name Enosh. Then began men to call upon the name of Jehovah.
Brenton Septuagint Translation
And Seth had a son, and he called his name Enos: he hoped to call on the name of the Lord God.
Contemporary English Version
Later, Seth had a son and named him Enosh. About this time people started worshiping the LORD.
Douay-Rheims Bible
But to Seth also was born a son, whom he called Enos; this man began to call upon the name of the Lord.
English Revised Version
And to Seth, to him also there was born a son; and he called his name Enosh: then began men to call upon the name of the LORD.
Good News Translation
Seth had a son whom he named Enosh. It was then that people began using the LORD's holy name in worship.
GOD'S WORD® Translation
A son was also born to Seth, and he named him Enosh. At that time people began to worship the LORD.
International Standard Version
Seth also fathered a son, whom he named Enosh. At that time, profaning the name of the LORD began.
JPS Tanakh 1917
And to Seth, to him also there was born a son; and he called his name Enosh; then began men to call upon the name of the LORD.
Literal Standard Version
And to Seth, to him also a son has been born, and he calls his name Enos; then a beginning was made of preaching in the Name of YHWH.
NET Bible
And a son was also born to Seth, whom he named Enosh. At that time people began to worship the LORD.
New Heart English Bible
And a son was also born to Seth, and he named him Enosh. This one began to call on the LORD's name.
World English Bible
There was also born a son to Seth, and he named him Enosh. Then men began to call on Yahweh's name.
Young's Literal Translation
And to Seth, to him also a son hath been born, and he calleth his name Enos; then a beginning was made of preaching in the name of Jehovah.