Before us is Mishnah 4 in the fourth chapter of Maseches Demai, which deals with the separation of terumos and maasros done on Friday afternoon, and with the question of what may and may not be done with them on Shabbos itself. Let us look at the wording of the Mishnah and its precision.
"Mi shekara shem li'terumas maaser shel demai u'l'maaser ani shel vadai":
The Mishnah speaks of a person who, on Friday afternoon, designated and gave a name to portions of the produce in his possession:
"Terumas maaser shel demai" - produce he bought from an am ha'aretz. He first separated maaser rishon from it, and from that maaser he separates the terumas maaser that must be given to a kohen.
"Maaser ani shel vadai" - maaser ani taken from certain tevel, that is, produce that was not bought from an am ha'aretz but came from a field that is clearly known to require the separation of maasros.
"Lo yitlem b'Shabbos":
Even though he gave them a name, he may not actually separate them and remove them on Shabbos. This phrase has two explanations:
Even though he designated the portions on Friday afternoon, he cannot actually separate them and remove them on Shabbos, since terumos and maasros are not separated on Shabbos.
And even if he separated them before Shabbos, he cannot give them to the kohen or to the poor man on Shabbos, since such giving is not permitted on Shabbos.
"Im hayah kohen o ani lemudim le'echol etzlo - yavo'u v'yochlu, u'vilvad she'yodi'em":
If the kohen or the poor man regularly eat at his home, they may come and eat. This is not giving a gift on Shabbos, but rather eating at his table as his guests normally do. There is, however, one condition: "U'vilvad she'yodi'em" - he must inform them that they are eating what is due to them by law, and not from the householder's own food - both the terumas maaser that goes to the kohen and the maaser ani that goes to the poor man. For otherwise, it turns out that he is feeding his guests from terumos and maasros that are not his, and that is forbidden.
Why did the Mishnah specifically speak of terumas maaser of demai?
After all, regarding maaser ani the Mishnah spoke of a case of certainty, of produce that is definitely obligated in the separation of maaser ani. Why, then, did it not also speak of terumas maaser of certain produce?
The reason is that the Mishnah is dealing with a Yisrael who bought demai and gives from it the gifts of the kehunah or the gifts of the poor. This person is a Yisrael and not a Levi, and seemingly he has no terumas maaser at all: terumas maaser under normal circumstances belongs to the Levi, and it is the Levi who must give it to the kohen. The only situation in which a Yisrael has terumas maaser to give is when he himself separates the maaser rishon and takes the terumas maaser out of it.
And that is precisely the case with demai: since the matter is in doubt, the Yisrael need not give the maaser rishon to the Levi, and therefore he himself is responsible for giving the terumas maaser to the kohen. This is why the Mishnah spoke of terumas maaser of demai, and not of terumas maaser of certain produce.
In summary: This Mishnah deals with someone who, on Friday afternoon, gave a name to terumas maaser of demai and to maaser ani of certain produce. We learned that they may not be separated and given on Shabbos, but if the kohen or the poor man regularly eat at his home, they may come and eat, provided that he informs them that they are eating what is due to them and not the householder's own food. And we noted the precision in the Mishnah's wording: terumas maaser was mentioned specifically in the case of demai, since only there does the responsibility fall upon the Yisrael himself to separate maaser rishon and to take the terumas maaser out of it for the kohen.