just my

One session may decide the situation you describe in one direction, one in another.
If it is felt the question falls closer to the distinction between a
regular and an
irregular administration of valid baptism, rather than between a
baptism and
no-baptism, the session may grant the lawfulness of the baptism as one that was
1) intended to be a baptism, 2) done in water, in the name of the Holy Trinity, with no further ceremony, and
3) was recognized thereafter by other identifiable Christians and churches as a valid mark.
The missing element:
baptized by a lawfully ordained minister, should not be slighted or regarded as a minor point. But it is not so exclusionary that without it no baptism could ever be regarded as legitimate. Otherwise, all extraordinary circumstances would absolutely prevent baptism. Otherwise, it is a capitulation to the position of Rome and other bodies, who teach that their ministers and churches alone confer baptism legitimately.
On the other side of the opinion, it may be the view of a session that more was missing from the
intention, or the
manner, or the ability to
recognize said baptism as a valid mark, besides the
absence of the minister. If so, they might present the case that they do not see that the minimum requirements for a valid baptism were fulfilled. It was an
invalid administration.
For instance, they might argue that the idea of private house-churches and self-ordained patriarchs are actually so contrary to the NT doctrine of baptism and of order in the churches that such an ordinance (called "baptism" by the practitioners) was more akin to the baptism of cults than Christian baptism. Coming
into the church and
under it's ministry and discipline (broadly not narrowly conceived) is
marked by baptism.
Finally, it would be good to go over such points for and against among the session alone prior to any meeting with prospective members, to gain a consensus. Then, it would be good to inquire of the candidate for membership about his understanding of what baptism is and does. And, it would be good to obtain the candidate's own testimony of how he regards the rite he underwent, and if he is able with a clear conscience to tie his ritual experience with the biblical teaching of baptism and sacramental theology of the Presbyterian church he wishes to join.