To get a conceptual point out of the way first. The Rachels' statement is ambiguous :
For its requirement of "exclusive" pursuit of self-interest is ambiguous. It may mean either (a)
that the egoist ought to do those actions of which he is the sole
beneficiary (thus he will be justified in taking those actions which will
result in a benefit to himself alone), or (b) that he ought to do only those
actions for which his motive is promotion of his interest (thus he is
justified in doing actions which will benefit himself along with others, but
his reason for acting must be to benefit himself). (Edward Regis, Jr, 'What is Ethical Egoism?', Ethics, Vol. 91, No. 1 (Oct., 1980), pp. 50-62 : 52.
Regis is not commenting on the Rachels' formulation of ethical egoism but on one precisely similar.
There is no need to tie ethical egoism to self-interest - an exclusive tie to interest, a person's interests, is sufficient.
But for the sake of argument, if we keep to 'self-interest', ethical egoism prescribes what one ought to do, what one should do, from a moral point of view.
'Psychological egoism' has nothing directly to do with ethics. It is a claim within psychology - rather old psychology - that people not only do pursue solely self-interest but that they cannot but pursue solely self-interest. Said another way (adapted from W.D. Glasgow : 75), it is a doctrine of motivation to the effect that human beings are so constituted that each seeks, and can only seek, self-interest .
Whereas ethical egoism prescribes, psychological egoism only describes deterministically.
Psychological egoism - indirect ethical implications
If 'ought implies can' then there is no point in holding that people ought to act any other way than solely from self-interest. The 19th-century utilitarians such as Jeremy Bentham and J.S, Mill had a way round this problem, however. Utilitarianism measures the moral value of actions purely by their consequences, not their intentions. If, then, you apply sanctions against morally undesirable behaviour, and the penalty imposed by the sanctions makes it in an agent's self-interest to desist from morally undesirable behaviour (judged by consequences) and it encourages morally desirable behaviour (again judged by consequences).
The truth is that the idea of a penal sanction, which is the essence of law,
enters not only into the conception of injustice, but into that of
any kind of wrong. We do not call anything wrong, unless we
mean to imply that a person ought to be punished in some way
or other for doing it; if not by law, by the opinion of his fellow-
creatures; if not by opinion, by the reproaches of his own con-
science. This seems the real turning point between morality and simple expediency. It is a part of the notion of Duty, in every
one of its forms, that a person may rightfully be compelled to
fulfill it. ( J. S. Mill, Utilitarianism, chap. 5.) in Utilitarianism, On Liberty, Essay on Bentham,
ed. M. Warnock (London: Fontana, 1962), pp. 303-4.
Note that connecting sanctions with self-interest in this way will not produce morality on a Kantian approach, where intention is central and consequences are not so.
Edward Regis, Jr, 'What is Ethical Egoism?', Ethics, Vol. 91, No. 1 (Oct., 1980), pp. 50-62.
W. D. Glasgow, 'Psychological Egoism', American Philosophical Quarterly, Vol. 13, No. 1 (Jan., 1976), pp. 75-79.
J.S. Mill, Utilitarianism in Utilitarianism, On Liberty, Essay on Bentham,
ed. M. Warnock (London: Fontana, 1962) or see : https://www.utilitarianism.com/mill1.htm.