The operative word, אוֹת (oth) used in Ex 31:13, Eze 20:12, 20, and correctly translated, "sign" is used of various things such as miracles, etc, see appendix below.
However, in all the instances quoted by the OP, the word is used in the technical sense of a sign/pledge of a covenant promise. In this case, the relevant covenant is the Israelite covenant as set out in Ex 19-23 involving the 10 commandments. This was distinct from the Levitical covenant (Lev 1-9, 16, 21-27 , Num 3, 4, 8, 18, 25:10-13, Deut 33:8-11, Neh 13:29, Mal 2:4-8).
All divine covenants had such "signs" such as:
- The Noahide covenant sign was the rainbow, Gen 9:12,13,17
- The Abrahamic covenant sign was circumcision, Gen 17:10, 11, 13
- The Israelite covenant involving the 10 commandments (Ex 34:27, 28, Deut 9:9, 11, 15) had as its central object the "Ark of the Covenant" (Num 10:33, 14:44, Deut 10:8, 31:9, 25, 26, Josh 3:3, 6, 8, 11, 14, 17, 4:7, 9, 18, 6:6, 8:33, etc) or, “Ark of the Testimony” (Ex 25:22, 26:33, 34, 30:6, 26, 31:7, 39:35, 40:3, 5, 21, Lev 16:13, Num 4:5, 7:89, Josh 4:16, etc) is described thus precisely because it contained the stone tablets inscribed by the finger of God (Ex 31:18, Deut 9:10) with the Israelite Covenant of the 10 Commandments. 1 Kings 8:9, 2 Chron 5:10.
[This is conformed by the introduction of the term in Ex 24:7, the “Book of the Covenant” as already existing, ie, Ex 20:22 – 23:33. It served as an expansion and legal context in which to place the Israelite Covenant of the 10 Commandments.]
The Sign of the this Israelite covenant was the Sabbath, Ex 31:13, Eze 20:12, 20, and usefully helped to distinguish Israel from the surrounding nations as because of what they kept.
NOTE: The extent to which Israel was alone in keeping, or at least recognizing the Sabbath is debated as numerous European peoples and their various languages have also preserved the same result. For example, observe the word for “Saturday” in each of the following languages which means “Sabbath”: Subbota (Morduin), Subota (Permian), Szombat (Hungarian), Sabbaton (Greek), Sabato (Italian), Sabado (Spanish), Sabbado (Portuguese), Samedi (French), Sobota (Polish), etc, etc. Numerous ancient peoples (more ancient than Jesus and the apostles) similarly preserve Saturday as the seventh day, eg, Coptic, Kabyle, Pashto, Persian, Armenian, Kurdish, Brahuiky, Turkish, Lazen, etc, etc.
APPENDIX - Brown-Driver-Briggs entry for אוֹת
אוֺת noun masculine79 Exodus 4:8 (feminineJoshua 24:17) sign (Arabic , plural , Aramaic אָתָא, ) — ׳א Genesis 4:15 +; construct Genesis 9:12 +; plural אוֺתוֺת Exodus 4:9 #NAME?
1 sign, pledge, token Genesis 4:15 (J); אוֺת אֱמֶת true, token Joshua 2:12 (J); of blood of passover Exodus 12:13 (P); אוֺת לְטוֺבָה token for good Psalm 86:17; pledges, assurances of travellers Job 21:29.
2 signs, omens promised by prophets as pledges of certain predicted events 1 Samuel 10:7,9 + 1 Samuel 10:1 where ׳א insert ᵐ5 ᵑ9, see We Dr; especially phrase זֶה הָאוֺת לְ Exodus 3:12; 1 Samuel 2:34; 1 Samuel 14:10; 2 Kings 19:29; Jeremiah 44:29; Isaiah 7:11,14; probably also Isaiah 44:25 (of false prophets).
3 sign, symbol of prophets Isaiah 8:18 compare Ezekiel 4:3.
4 signs, miracles, as pledges or attestations of divine presence & interposition Exodus 4:8 (twice in verse); Exodus 4:9 (J) Exodus 7:3 (P) Exodus 8:19 (J) Psalm 74:9; 2 Kings 19:29; 2 Kings 20:8,9 = Isaiah 37:30; Isaiah 38:7,22; with עָשָׂה Exodus 4:17,30; Numbers 14:11,22; Joshua 24:17 (all J E) Deuteronomy 11:3; Judges 6:17; with צִוָּה Exodus 4:28 (J); with שִׁית Exodus 10:1 (J); with שִׂים Exodus 10:2 (J) Psalm 78:43; Isaiah 66:19; הָאוֺת וְהַמּוֺפֵת (see מוֺפֵת) Deuteronomy 13:3; compare Deuteronomy 28:46; Isaiah 20:3; ׳אוֺ מ ׳א Deuteronomy 13:2; אוֺתוֺת וּמוֺפְתִים Deuteronomy 4:34; Deuteronomy 7:19; Deuteronomy 26:8; Deuteronomy 29:2; Jeremiah 32:31; with נָתַן Deuteronomy 6:22; Nehemiah 9:10; with שִׂים Jeremiah 32:20; Psalm 105:27; with שָׁלַח Deuteronomy 34:11; Psalm 135:9.
5 signs, memorials, stones from Jordan Joshua 4:6 (J); metal of censers Numbers 17:3 (P); Aaron's rod Numbers 17:25 (P); עוֺלָם ׳א Isaiah 55:13 probably also Ezekiel 14:8 ("" מָשָׁל); signs on hands, etc., Exodus 13:9,16 (J) = Deuteronomy 6:8; Deuteronomy 11:18, probably belong here; also memorial pillar in Egypt Isaiah 19:20.
6 sign, pledge of convenant, הַבְּרִית ׳א (see ברית) e.g. rainbow, of Noachian convenant Genesis 9:12,13,17 (P); circumcision, of Abrahamic convenant Genesis 17:11 (P); the sabbath Exodus 31:13,17 (P); Ezekiel 20:12,20.
7 ensigns, standards Numbers 2:2 (P) Psalm 74:4.
8 signs, tokens of changes of weather & times Genesis 1:14 (P; of heavenly luminaries) אוֺתוֺת הַשָּׁמַיִם Jeremiah 10:2 (changes of the heavens as omens to frighten the nations) compare Psalm 65:9.