Date of the first Eclipse in the past

Hi Mr. Wilhelm and friends,

I was analyzing the upcoming Solar Eclipse on the 14th of October 2023. From the "Eagle and the Lark" book I have taken notes for which one the Saros Cycles the upcoming transits belong to. This one is South 7. And in the book it gives the First Eclipse to be born at June 22, 1248. When I create the chart in Kala it doesn't give the eclipse moment, it's just another day. 

Luckily, now I've just noticed your most recent video about the upcoming Eclipse! Thank you very much for that. I will watch it immediately, right after my question. 

In the first minutes, you are giving this eclipse to be born at June 29th, 1248, 13:49:33 Washington DC. And that one isn't an eclipse chart either. (In Kala the eclipse happens for Washington DC at May 31st, 1248, 06:54:55)

Saros South-7 Eclipse

Upcoming: October 14th, 2023

First Eclipse (Eagle and Lark): June 22nd, 1248

First Eclipse (Ernst Wilhelm Youtube): June 29th, 1248

First Eclipse (at Kala created): May 31st, 1248 

You mention that the calendar should be Gregorian, not Julian. As the Gregorian date was introduced in 1582, then should we convert any date before that year to Gregorian or are the settings already doing that?

In the end;

*What I would really like to know, how do you find the First eclipse date of a Saros Cycle more practically?

**Why there is a difference with the date you give for the First Eclipse (June 29th, 1248) and the one in Kala (May 31st,1248), although my settings are default as recommended?

I would be happy to know and understand if you could help me.

Thank you

Zumrut