It can be helpful to consider a marriage at the moment of the exchange of consent—i.e., when it comes into existence—and distinguish that from the marriage as it endures through time.
The divisions that you refer to all have to do with whether the necessary conditions are in place in order for the marriage to come into existence in the first place; that is, they have to do with the exchange of consent.
Canonical form, disparity of cult, and mixed marriage
Regarding this, the Church essentially imposes two conditions for the validity of a marriage, which go beyond what the natural law requries: namely, (1) that the couple exchange their vows before a properly delegated witness (also known as the requirement of “canonical form;” see Code of Canon Law [CIC] 1108); and (2) that (assuming that at least one party is Catholic) both parties be baptized. (See CIC 1086.) Failure to comply with these conditions is an “impediment” to marriage: if a couple attempts to marry without complying, the marriage is invalid; that is, it never comes into existence.
(There are actually more impediments, but those two are the most commonly dealt with. Many other impediments can be found in the CIC 1083-1094. Note that these impediments generally apply only when one or both of the parties are Catholic; two non-Catholics are not bound by these requirements.)
Both of the requriements can be dispensed with by the local ordinary of the Catholic party (CIC 1078). Evidently, the second condition can only occur when one party is Catholic and the other is non-Christian, and the first condition ordinarly becomes an issue only when a Catholic wants to marry a non-Catholic.
A Catholic must request permission (licence) to marry a baptized non-Catholic; however, that permission is not necessary for the validity (existence) of the marriage, unlike the dispensations that I mentioned earlier. (See CIC 1124. Note that the text of this canon was changed in 2009.)
Requests for dispensations and permissions are part of the routine paperwork for marriage preparation. When a Catholic marries a non-Catholic, as the O.P. says, the Catholic party must declare that he will remove dangers of defecting from the Church, and must do all in his power (evidently, all that is reasonably in his power) to make sure that the childred are baptized and raised Catholic. There is actually no particular requirement to try to “convert” the non-Catholic spouse, nor must the non-Catholic party specifically promise not to prevent the Catholic party from practicing the faith (although doing so would seem to pose a “danger of defection.”) The conditions are spelled out in CIC 1125.
Note that as far as Church law is concerned, a person who has become a member of the Catholic Church is always considered a member of the Church for the purpose of marriage law, regardless of whether he practices or not, or whether he has joined another church. (See Benedict XVI’s motu proprio from 2009, which addresses this issue.) Likewise, there is no disparity of cult so long as the non-Catholic party is baptized (even if he does not practice any Christian religion). Hence, Church law does not have a special case for baptized non-Christians; they are treated like any other baptized non-Catholics.
(The baptism, however, must be one that is recognized by the Catholic Church as valid: specifically, it must be documented, and it must have been administered using the Trinitiarian formula, with at least the generic intention to do what the Church intends. If there is any doubt, the dispensation for disparity of cult should be sought.)
Sacramental vs. natural marriages
As I mentioned, these divisions have to do with how marriages come about. I think it would be a mistake to think of them as three different “kinds” of marriage. Rather, these distinctions describe three potential obstacles that might need to be overcome in order for a marriage to take place: two common impediments that need to be dispensed with (sometimes both at the same time), and one permission that might need to be requested.
If marriage is looked at as a permanent reality that endures in time, then in that case the Church does distinguish between two (not three) different kinds of marriages: natural marriages (those contracted when one or both of the parties is not baptized) and sacramental marriages (those contracted between two baptized persons, which automatically constitutes one of the seven Sacraments). This is explained very well in CIC 1055.