To answer questions about "stop words" such as עד, it is best to look directly at examples from the primary text, and not to attempt to understand from what secondary sources such as BDB, or tertiary source have to say. There are 250+ usages of עד in the OT with the meaning of "until", "to", "up to", for conditions or ranges. So we have many examples that we can study in order to elucidate the meaning.
WRT time or conditions, עד always indicates including. Therefore, in the instance of the jubilee year, additional words are necessary to qualify עד in order to express the meaning "up to but not including".
The proximal example to Exodus 12:6 is Exodus 12:10 (NIV):
Do not leave any of it till morning; if some is left till morning, you must burn it.
From this example it is clear that עד is inclusive of the morning.
So, Exodus 12:6 means including the 14th day of the month of spring (Aviv) as there are no further words or verses to indicate that the guarding of the lamb was to stop at the end of the previous day, the 13th day of the month.
A futher example, in Genesis 26:33, עד means including this day (NIV):
He called it Shibah, and to this day the name of the town has been Beersheba.
Same for Genesis 27:45:
When your brother is no longer angry with you and forgets what you did to him, I’ll send word for you to come back from there. Why should I lose both of you in one day?
Same for Genesis 28:15:
I am with you and will watch over you wherever you go, and I will bring you back to this land. I will not leave you until I have done what I have promised you.
Same for Genesis 29:8:
“We can’t,” they replied, “until all the flocks are gathered and the stone has been rolled away from the mouth of the well. Then we will water the sheep.”
Same for Genesis 32:25:
So Jacob was left alone, and a man wrestled with him till daybreak.
Same for Genesis 32:32:
Therefore to this day the Israelites do not eat the tendon attached to the socket of the hip, because the socket of Jacob’s hip was touched near the tendon.
Same for Exodus 10:6:
They will fill your houses and those of all your officials and all the Egyptians—something neither your parents nor your ancestors have ever seen from the day they settled in this land till now.’” Then Moses turned and left Pharaoh.
Same for ... (about 190 more)
In the instance of the Jubilee, Leviticus 25:28 additional words are required to quality עד to mean "not including:
But if they do not acquire the means to repay, what was sold will remain in the possession of the buyer until the Year of Jubilee. It will be returned in the Jubilee, and they can then go back to their property.