Is it true that the president can vote only to break a tie?

No, it is not true that the president can vote only to break a tie. If the president is a member of the voting body, he or she has exactly the same rights and privileges as all other members have, including the right to make motions, to speak in debate, and to vote on all questions. So, in meetings of a small board (where there are not more than about a dozen board members present), and in meetings of a committee, the presiding officer may exercise these rights and privileges as fully as any other member. However, the impartiality required of the presiding officer of any other type of assembly (especially a large one) precludes exercising the rights to make motions or speak in debate while presiding, and also requires refraining from voting except (i) when the vote is by ballot, or (ii) whenever his or her vote will affect the result.

When will the chair's vote affect the result? On a vote that is not by ballot, if a majority vote is required and there is a tie, he or she may vote in the affirmative to cause the motion to prevail. If there is one more in the affirmative than in the negative, the chair can create a tie by voting in the negative to cause the motion to fail. Similarly, if a two-thirds vote is required, he or she may vote either to cause, or to block, attainment of the necessary two thirds. [RONR (11th ed.), pp. 405-6; see also Table A, p. 190 of RONRIB.]