The short answer is a Hebrew is someone descended from Abraham, an Israelite is someone descended from Jacob, and a Jew is someone descended from Judah or someone whose cultural heritage is from the Kingdom of Judah. No they are not all the same.
The long answer delves deep into the Old Testament and also into more "modern" times and relies on the study of linguistics.
The fist time we see the word "Hebrew" in the Bible is in Genesis 14:13 where we are told
The word used in this verse is הָעִבְרִ֑י (hā·‘iḇ·rî) and it translates to "the Hebrew". The word for Hebrew is עִבְרִ֖י (‘iḇ·rî), please note the lack of הָ (hā) "the" as this is very important in a different discussion. To understand what the word Hebrew meant in Genesis 14:13 we need to look at the history of Abraham. Abraham was originally named Abram and he was originally from the city of Ur which was Chaldeean. Abram at the time of Genesis 14:13 was living in Canaan, he was in effect a foreigner which is what he was called. Abram the Hebrew means Abram the foreigner or more correctly Abram from the other side. Other side of what? I would suggest a territorial border others suggest the Euphrates River because after he left Ur Abram travelled to Harran on the eastern side of the Euphrates and stayed there. Ur was west of the Euphrates, Harran on the East, and Canaan where Abram was in Genesis 14:13 is west of the Euphrates but also west of the Jordan River. So the word Hebrew is an epithet used to describe Abram as a foreigner in Canaan.Then a fugitive came and told Abram the Hebrew. Now he was living by the oaks of Mamre the Amorite, brother of Eshcol and brother of Aner, and these were allies with Abram.
Abram is also called an Aramean in Deuteronomy 26:5. An Aramean is someone from Aram. Aram is the Hebrew word for Mesopotamia (Mesopotamia simply means "land between 2 rivers"). Ur of the Chaldees on the western side of the Euphrates and is where Abram come from. There was a Kingdom of Aram in the area we now know as Syria but there was also a "tribal" area of Aramean tribes west of the Euphrates River with another Aram (Aram Naharaim) between the Euphrates and Tigris Rivers. So Abram is both a Hebrew and an Aramean. It is also worth noting that some parts of the Bible are written in Aramaic.
Remember I said above to note the lack of הָ (hā) "the". Many people suggest the people we now call Hebrew are the same people who are written about in ancient documents like the Amarna letters and called Hapiru or Apiru. The word Hebrew is a modern translation of the word (written in ancient Hebrew) ibri (or ivri). If we consider the Amarna letters were not written in Hebrew but rather in Akkadian cuneiform (which like Hebrew and Aramaic are Semitic languages) to the Egyptian Pharoah if the 'ha" or "a" was taken off the apiru/hapiru we would be left with "piru". Piru does not look like or sound like "ibri or ivri" In a letter from Rib-Addi of Byblos to the Egyptian Pharoah Rib-Addi says
and"However, the war of the 'Apiru against me is severe."
"Be informed that since Amanappa's arrival to me, all the 'Apiru have turned their face against me at the instigation of Abdi-Ashirta."
"And if there are no archers, then all the lands will unite with the 'Apiru. Listen, since the conquest of Bit-Arha in accordance with the demand of Abdi-Ashirta, they seek in the same way to unite Gubla and Batruna; and thus all lands would be united with the 'Apiru."
Now if haibri (or haivri) translates to "the Hebrew" or "the ibri", apiru and hapiru (being the same word) cannot be the equivalent of Hebrew (ibri) because the text would then read "the the Hebrew" or "the the ibri". So the apiru/hapiru cannot be considered to be Hebrews (ibri) based on vague similarities in the sounds of the words in todays modern languages.
The word Hebrew (ibri or ivri) that we use when reading the Bible is not used very often and the times it is used it is used as an identifier of the Hebrew people to non-Hebrew people or as an identifier of Hebrew people by non-Hebrew people. The word apiru/hapiru literally means nomadic or semi-nomadic, rebels, outlaws, raiders, mercenaries, bowmen, servants, slaves, migrant labourer. Considering the word apiru/hapiru is used by people who were subservient to Egypt in what is now the middle east west of Iran the letters would most likely have referred to the escaped slaves mentioned in Exodus as being the same people if they were Hebrews (ibri) but the letters make no mention of this. Why? I suggest because they are not the same people but rather the word apiru/hapiru is a generic word that labels any foreigner, hostile group, or slave so the locals have a way to identify them as non-desirables (a label used in early class warfare if you will).
The first we read of Israel is Genesis 32:28 where Elohim (the God) renames Jacob Israel because he "has struggle with God and man and prevailed".
To be continued.....