Yes, the definite article is specifically referring to the days God had defined, and prophesied for the end of the Mosaic sacrificial covenant, and the end of last of the tribes of Israel - Judah. In defining words and phrases used in the NT, it is best to search for their first use in the OT. The NT is the OT revealed.
The Complete Jewish Bible (CJB) and the Orthodox Jewish Bible (OJB) translate the Greek with the definite article "the" before using Hebrew "acharit-hayamim" meaning perilous times and is linked to the "end of days" of Daniel 12:13.
The placement of the article either before or after, or implied is rendered the same in all of the English translations I have found for each of these verses. The understanding of the translators appears to agree in all of them as well as in the Interlinear and Young's, that it is "the last days."
The first use of "the last days," or "the latter days" is found in Jacob's prophesies in the blessings given to his sons before he died.
"And Jacob called unto his sons, and said, Gather yourselves together, that I may tell you that which shall befall you in the last days." (Gen. 49:1, KJV)
Jacob was not prophesying the end of time, nor the end of the natural, physical world. He was telling his sons how each of their line (or tribe) would end. The last days or the latter days was about the last days of the tribes of Israel. It was their last days, not the last days of time.
Jacob detailed the circumstances of each of his line, but told them that Judah would remain until the "Seed" or "Shiloh" came.
"The sceptre shall not depart from Judah, nor a lawgiver from between his feet, until Shiloh come; and unto him shall the gathering of the people be." (Gen. 49:10, KKV)
Tying the last days of Judah to the manifestation of Christ defines “the last days” as the time in which the law passed from Judah to Christ which happened at the cross of Christ. Therefore the last days were the days in which Christ was manifested on earth during the first century A.D, and in which He was crucified.
The phrase next appears in Num. 24:14-18 where Balaam gave a prophesy to Balak.
14 And now, behold, I am going to my people; come, I will let you know what this people [the Israelite] will do to your people [the Moabite] in the latter days.” 15 And he took up his discourse, and said,
“The oracle of Balaam the son of Be′or,
the oracle of the man whose eye is opened,
16 the oracle of him who hears the words of God,
and knows the knowledge of the Most High,
who sees the vision of the Almighty,
falling down, but having his eyes uncovered:
17 I see him, but not now;
I behold him, but not nigh:
a star shall come forth out of Jacob,
and a scepter shall rise out of Israel;
it shall crush the forehead[c] of Moab,
and break down all the sons of Sheth.
18 Edom shall be dispossessed,
Se′ir also, his enemies, shall be dispossessed,
while Israel does valiantly.
19 By Jacob shall dominion be exercised,
and the survivors of cities be destroyed!”
(Num. 24:14-18, RSV)
The last days were again defined as when "a star shall come forth out of Jacob" who will have dominion - Christ.
The definite article "the" was calling to mind the prophesies God had given concerning the Israelite people, and the time when Messiah would come.
Beginning in Acts 2:17, following the colon in vs. 16, Peter repeats Joel's prophesy of things that would happen in "the last days."
"28 And it shall come to pass afterward, that I will pour out my spirit upon all flesh; and your sons and your daughters shall prophesy, your old men shall dream dreams, your young men shall see visions: 29 And also upon the servants and upon the handmaids in those days will I pour out my spirit." (Joel 2:28-29, RSV)
Joel had prophesied of the pouring out of the Holy Spirit upon the people that would happen before "the great and terrible day of the Lord comes" (Joel 2:31) which was spoken against Jerusalem and Judea (Joel 2:32), and only those called would escape that judgment.
Peter told the people standing at the temple in Jerusalem who were witnessing the miracles on the day of Pentecost -
"15 For these men are not drunk, as you suppose, since it is only the third hour of the day; 16 but this is what was spoken by the prophet Joel:" (Acts 2:15-16, RSV)
Peter was stating absolutely that the miracles they witnessed on the day of Pentecost was that pouring out of the Holy Spirit which would happen in "the last days". As they were witnessing the fulfillment of the prophesy in 30-31 AD, Peter affirmed his time as being "the last days" in the first century A.D.
The last days were from the time of the manifestation of Christ (1 Pet. 1:20 "these last times") to the destruction of Jerusalem and the temple AD 70 which were the last days of the tribe of Judah and the temple worship system of animal sacrifices in Jerusalem.
Hebrews 1:1-3 affirms the time again for "these last days" as the days in which Christ spoke directly to the people, the days in which He was crucified. The demonstrative pronoun "these" defined their days of the first century AD, their days in which the book of Hebrews was written.
"These" days are not our days when we are reading them almost 2,000 years later. "These days" were those days, their days in which the words were spoken to the people of the first century A.D. First audience perspective.
"The last days" were defined in the OT as the last days of the tribes of Israel, and as Judah was the one who would remain as the "lawgiver" until the scepter passed to Christ, then the last days were the time when Christ was crucified and given dominion over all the earth. His dominion began at the cross, and was finalized after the spread of the gospel throughout the inhabited world (Matt. 24:14) when He returned to destroy the temple and Jerusalem in AD 70, after which He was given dominion to reign forever at the right hand of the Father (Dan. 7:22-27).
The use of the definite article "the" is critical, and has been inserted at other times when it is not in the original text (Heb. 9:27). However, in these verses the definite article "the" was intended and was very specific.
The translations of Heb. 1:2 is interesting in that many do express the meaning of "end of days" for "these last days." For example,
ASV: "hath at the end of these days"
CEB: "In these final days"
CJB: "But now, in the acharit-hayamim"
DLNT: "spoke to us at the last of these days"
ERV: "And now in these last days"
PHILLIPS: "has now, at the end of the present age"
NTE: "but at the end of these days "
OJB: "At the Ketz HaYamim,"
The variations stress more completely the then current recognition that those of the 1st century AD were living in the "last days" or the "end of days" of the Mosaic age.
On Hebrews 1:1-3, see "It's Not The End of The World - Part I..." ShreddingTheVeil
On the inhabited world of Matt. 24:14 see "Frequent Mistakes - Part IV: Where was All the World" ShreddingTheVeil
More scriptural evidences - "Frequent Mistakes - Part III: The Last Day" ShreddingTheVeil
More on Dan. ch. 7 - "Frequent Mistakes - Part II: The Ascension of Christ" ShreddingTheVeil