The Torah commands us: "In the First Month on the fourteenth day of the month, between the two evenings [Hebrew: "Bein Ha'arabayim"], is the Passover [Sacrifice] to Yehovah." In biblical Hebrew, the word "evening" (Ayin-Resh-Bet) indicates both the "early part of the night", as well as the actual "onset of evening". In the expression "between the two evenings" the first "onset of evening" is sunset (when the disk of the sun disappears) while the second "onset of evening" is the disappearance of the last rays of the sun and the onset of total darkness. The expression "between the two evenings" is used interchangeably with the term "Ba-Erev" (literally: "at evening") which itself refers to the "onset of the evening".
For example, in the incident of the Manna it is written (Exodus 16:11-13):
"I have heard the complaints of the Children of Israel; speak to them saying 'Between the two evenings you shall eat meat'... And it was at evening that the quail rose up and covered the camp.'
We see in this passage that an event predicted as happening "between the two evenings" is said to have happened "at evening". The meaning of "at evening" itself can be learned from the verse "... you shall slaughter the Passover [sacrifice] at evening, at sunset" (Deuteronomy 16:6). We see in this verse that "at evening" and "at sunset" are interchangeable expressions (used in "apposition").
To summarize, the Torah describes the time of the Passover Sacrifice with three different expressions: "At Sunset", "At Evening", "Between the Two Evenings". All three of these terms refer to the early evening, shortly after sunset.
Beginning or End of the 14th?
The Torah commands that the Passover sacrifice be brought "In the First Month on the Fourteenth Day of the Month between the two evenings" (Leviticus 23:5). It is unclear from this verse whether what is being referred to is the period of dusk at the beginning of the 14th or the period of dusk at the end of the 14th. Leviticus 23:6 continues that the Feast of Unleavened Bread is "on the Fifteenth Day of this month". From this verse it appears that the Passover Sacrifice is to be brought at sunset at the end of the 14th, and eaten that same evening, on the night of the 15th. This is confirmed by Deuteronomy 16:4, which commands us regarding the Passover Sacrifice: "and there shall not remain of the meat that you slaughter at evening on the first day until the morning." We see that the entire Paschal lamb must be consumed on the following night it is slaughtered and none of it may be left over until the morning (see also Exodus 12:10, 22). For our purposes what is significant is that the verse describes the Passover sacrifice as being slaughtered "at evening on the first day".
The passage in Deuteronomy 16:1-8 is talking about the Feast of Unleavened Bread and there can be no doubt that "the first day" in v.4 refers to the first day of Unleavened Bread. We have already seen in Leviticus 23:6 that the First Day of Unleavened Bread falls out on the 15th of the First Month. When we look at Leviticus 23:5-6 and Deuteronomy 16:4 together it becomes clear that the Passover Sacrifice is brought at the end of the 14th of the First Month between the two evenings and eaten that same evening on the 15th of the First Month. The period of "between the two evenings" is reckoned as both the end of the 14th (Leviticus 23:5) and the beginning of the 15th (Deuteronomy 16:4)!
It is not unusual for the Torah to refer to "such and such a date at evening" and to mean the evening that ends that day. In Leviticus 23:27 we learn that the Day of Atonement occurs on the 10th day of the Seventh Month. A few verses later the Torah makes clear what is meant by the 10th day: "and you shall afflict your souls on the ninth of the month at evening, from evening to evening you shall observe your Sabbath" (Leviticus 23:32). So we see that to fast on the 10th day means to fast from sunset on the 9th until the following sunset. In this verse "the ninth at evening" refers to the onset of evening at the end of the 9th, not the beginning! So the fast of the Day of Atonement on the 10th of the month runs from sunset ending the 9th until sunset ending the 10th (see also Exodus 12:18). Similarly, the 14th between the two evenings in verse 5 of the same chapter refers to the end of the 14th, not the beginning, as confirmed by Deuteronomy 16:4.
"and there shall not remain of the meat that you slaughter at evening on the first day until morning"