When they are occupied with the same matter, the witnesses know what is happening, but in silence it could be that he is buying something, repaying a debt, or giving a gift. So how do the witnesses know that this is betrothal?

When it is clear from the context that they intend betrothal, or when the couple told them that it was being given for the sake of betrothal.

I understood that there are three approaches regarding verbal declaration in betrothal. One approach is that the declaration itself effects the betrothal. A second approach is that the declaration is not essential, but rather is needed so that the betrothal not remain merely things in one’s heart; therefore, if it is clear from the context, or if someone was told that this was being given for betrothal, then even without a declaration the betrothal takes effect. And a third approach is that of the Mordechai, that even a declaration to remove it from the category of things in one’s heart is not needed. According to the Mordechai’s approach, how can one testify to the betrothal?

I don’t understand what seems problematic to you in what I wrote. Obviously there has to be intent to effect betrothal. Nobody gives that up.

Obviously intent is required, but according to the Mordechai it does not need to be expressed outwardly. The Rabbi wrote that one can testify to the betrothal when it is clear from the context or when the couple told the witnesses. But according to the Mordechai, even that is not necessary. The couple’s intent alone is enough. If it is clear from the context or they told the witnesses, that is the second approach I mentioned, not the Mordechai. In the case the Mordechai is talking about, how can the witnesses testify to the betrothal if the couple’s intention is not clear (even though the couple know what they intended), assuming the Mordechai is disagreeing in a real case and not just saying theoretically that one could effect betrothal silently if the witnesses knew how to read minds.
Now I was thinking that maybe there could be some particular sign that only the witnesses recognize, showing that this is betrothal, and then it would still be silent betrothal that would work only according to the Mordechai and not according to those who disagree with him. Does that seem like a good solution to the Rabbi?

I don’t know what that sign would be. I find it hard to believe that the Mordechai means a case where there is no indication at all from the circumstances that the intent is betrothal. However, from the plain wording of the Rema, section 27 subsection 3, it does appear that indeed no sign at all is needed, and the couple’s admission is enough. Perhaps his intention is to say that if both of them admit it, then this is a litigant’s admission, and it is considered like two witnesses even for establishing the matter itself (and the dispute between Ketzot HaChoshen and the Rashba on this is well known).

What I meant was, say the witnesses are friends of the man and know that he touches his hair every time he effects betrothal; then only the Mordechai would say that the betrothal takes effect, because those who disagree with him would say that this is just things in one’s heart (since touching one’s hair is not a normal sign of betrothal, and it is clear only to his friends). But the Rabbi’s solution is better, because it doesn’t seem plausible to me that the Mordechai is disagreeing in such a case. Thank you.