Praise be to Allah, and peace and blessings be upon the Messenger of Allah. To proceed:
This issue was discussed by the scholars of Islam from early times. The answer depends on what is meant by the word "believer" (mu'min).
If a person means the basic faith by which one leaves disbelief and enters Islam, then they should not say, "I am a believer, if Allah wills." In this case, such an expression may be understood as doubting the foundation of one's faith, whereas a Muslim should be certain of their Islam and faith in this sense.
Therefore, they should say:
However, if one means the perfection and completeness of faith—that they have reached the highest level of faith deserving praise—then it is better to say:
"I am a believer, if Allah wills."
This is out of humility before Allah, fear of praising oneself, and in accordance with the saying of Allah:
"So do not claim yourselves to be pure; He knows best who is truly God-fearing."
(Qur'an 53:32)
If a Muslim wants to use an expression that avoids any misunderstanding, the best wording is:
"I am a Muslim."
This is a correct expression that requires no qualification, does not imply doubt, and does not involve praising oneself.
If the intention is the foundation of faith and Islam, then one should not say, "I am a believer, if Allah wills," in a way that suggests doubt. Rather, they should affirm their faith. However, if the intention is the perfection of faith and humility before Allah, there is no harm in saying, "I am a believer, if Allah wills." The safest expression is to say:
"I am a Muslim."
And Allah knows best.