It means the people of God.

The Israelite people (not jews).

In Galatians, Paul was writing to the Israelite people whose ancestors had lived in the Kingdom of Israel, then taken captive by the Assyrians, and later moved north.

Galatia was the area that is today known as western Turkey.

The people living there during Paul's time were decended from the original Israelites.

They had already started drifting away from their heritage and not living under The Law or following Christ.

He was writing the epistle (letter) to them, to remind them how they were supposed to live their lives.

Galatians 1:1-4 --

1 Paul, an apostle, (not of men, neither by man, but by Jesus Christ, and God the Father, who raised him from the dead;

2 And all the brethren which are with me, unto the churches of Galatia:

"Bretheren" because Paul was also an Israelite. They were genetic brothers, related by blood.

3 Grace be to you and peace from God the Father, and from our Lord Jesus Christ,

4 Who gave himself for our sins, that he might deliver us from this present evil world, according to the will of God and our Father:

"Our sins," "might deliver us, "our Father."

Galatians 5:1 --

Stand fast therefore in the liberty wherewith Christ hath made us free, and be not entangled again with the yoke of bondage.

Who had been in bondage? These Israelite people who were now living in Galatia, or their parents, grandparents, and other immediate ancestors.

In Galatians 6, Paul is talking about the practice of circumcision to his brothers. In the Old Testament, God wanted His men to be circumcised, probably to distinguish them from other people who were not His people.

But since those old days, the jews (not God's people) had adopted the practice of circumcision, too, and used that as a basis to claim they were of God's people.

Here, Paul is saying that Jesus/God no longer cares about circumcision, because it is no longer a distinguishing characteristic of His people. The jews had "culturally approprated" the practice, so to speak, so it had lost its meaning.

The passages you cited:

15 For in Christ Jesus neither circumcision availeth any thing, nor uncircumcision, but a new creature.

Circumcision was no longer a requirement of God's people, and instead following Christ is what God expected of His people ("a new creature," meaning a change in understanding The Law, since times had changed; that is why God came to Earth in human form, calling himself Jesus).

16 And as many as walk according to this rule, peace be on them, and mercy, and upon the Israel of God.

God now wants His people to walk in the way that Jesus taught, and if they do, then mercy shall be upon the people of God.

Seems like there's an Israel NOT of God

No, you are just not interpreting it in the proper context.

Modern churches do not teach the truth of the Bible, in many important aspects.

It means the people of God.

The Israelite people (not jews).

In Galatians, Paul was writing to the Israelite people whose ancestors had lived in the Kingdom of Israel, then taken captive by the Assyrians, and later moved north.

Galatia was the area that is today known as western Turkey.

The people living there during Paul's time were decended from the original Israelites.

They had already started drifting away from their heritage and not living under The Law or following Christ.

He was writing the epistle (letter) to them, to remind them how they were supposed to live their lives.

Galatians 1:1-4 --

1 Paul, an apostle, (not of men, neither by man, but by Jesus Christ, and God the Father, who raised him from the dead;

2 And all the brethren which are with me, unto the churches of Galatia:

"Bretheren" because Paul was also an Israelite. They were genetic brothers, related by blood.

3 Grace be to you and peace from God the Father, and from our Lord Jesus Christ,

4 Who gave himself for our sins, that he might deliver us from this present evil world, according to the will of God and our Father:

"Our sins," "might deliver us, "our Father."

Galatians 5:1 --

Stand fast therefore in the liberty wherewith Christ hath made us free, and be not entangled again with the yoke of bondage.

Who had been in bondage? These Israelite people who were not living in Galatia, or their parents and other immediate ancestors.

In Galatians 6, Paul is talking about the practice of circumcision to his brothers. In the Old Testament, God wanted His men to be circumcised, probably to distinguish them from other people who were not His people.

But since those old days, the jews (not God's people) had adopted the practice of circumcision, too, and used that as a basis to claim they were of God's people.

Here, Paul is saying that Jesus/God no longer cares about circumcision, because it is no longer a distinguishing characteristic of His people. The jews had "culturally approprated" the practice, so to speak, so it had lost its meaning.

The passages you cited:

15 For in Christ Jesus neither circumcision availeth any thing, nor uncircumcision, but a new creature.

Circumcision was no longer a requirement of God's people, and instead following Christ is what God expected of His people ("a new creature," meaning a change in understanding The Law, since times had changed; that is why God came to Earth in human form, calling himself Jesus).

16 And as many as walk according to this rule, peace be on them, and mercy, and upon the Israel of God.

God now wants His people to walk in the way that Jesus taught, and if they do, then mercy shall be upon the people of God.

Seems like there's an Israel NOT of God

No, you are just not interpreting it in the proper context.

Modern churches do not teach the truth of the Bible, in many important aspects.