Your Implication is not matching the given Statement. It seems to state that who-ever did the mailing & the telephoning must be $x,y$ , which may not be true when there are many such Students or Pairs of Students.
Consider these instead :
$$\exists x (\forall z((M(x,z) \lor T(x,z))))\tag{1}$$
This says that there is at least 1 student $x$ ( though there may be more ) who did the mailing to very student $z$ or did the telephoning to $z$.
We want at least 2 such students , hence this will work :
$$\exists x \exists y ((x≠y) \land \forall z((M(x,z) \lor T(x,z)) \land (M(y,z) \lor T(y,z))))\tag{2}$$
This makes the same claim about $x$ & $y$ & then it adds the Criteria that these two are not the SAME , hence we have at least 2 such students ( though there may be more ) matching the given Statement.
$$\exists x \exists y ((x≠y) \land \forall z((M(x,z) \lor T(x,z)) \lor (M(y,z) \lor T(y,z))))\tag{3}$$
This makes the same claim every Student has got Communication from either $x$ or $y$ , either by mail or by telephone & then it adds the Criteria that these two are not the SAME , hence we have at least 2 such students ( though there may be more ) matching the given Statement.
[[ I am adding (3) because I see "...who between them..." , which means both of them "Combined together" have Communicated with the other Students. ]]
Do not use Implication here , It is unnecessary here , It is enough to use $M()$ & $T()$ in this Case.
DIAGRAMMATIC EXPLANTION :
Consider 6 Students , $1\cdots6$ , & let the Black lines indicate Communication (Either Email or telephone) where the Arrow Directions indicate who contacted whom.
1 contacted : 2,4,6
2 contacted : 6
3 contacted : 2,4
4 contacted : 1,3
5 contacted : 4,6
6 contacted : 5
Which-ever 2 we take , we do not get all other Students.
Let the Blue Dashed line be a new email contact from 3 to 5.
Then (1,3) contacted 2,4,6 + 5 which is all other Students.
Is this what we want ? Question in English is ambiguous : It may be that 1 & 3 must also have been contacted. Then let the Purple Dashed lines be the new contacts between 1 & 3.
We can then state that there are 2 Students (1,3) who between them have contacted every other Student.
OK , let the Green Dashed lines be the new contact between 1 & 5.
Then 1 contacted : 2,3,4,5,6
5 contacted : 1,4,6
We can then state that there are 2 Students (1,5) who between them have contacted every other Student.
It is not unique , then we can not make the implication that which-ever Pair has this Property must be (1,3) , that Pair could be (1,5) too.
When more contacts occur , there may be more such Pair Choices.
In the Extreme Case where Every Student has contacted all Students , Every Pair is valid Choice !
Without Implication , we can simply state that there are 2 Students who have the Property , leaving it open whether that Pair is unique or not.