I don't think it has a name per se, but it definitely falls under the category of FreeFOP, where F2L isn't solved by rigidly doing a full cross then F2L pairs. It's not terrible.
Some awkward F2L cases have nicer solutions when you don't have to fuss over preserving a cross edge, e.g. r U r' U r U' r' instead of R2 U R2 U R2 U2 R2 or R2 U2 R2 U' R2 U' R2 (7 moves, but many wrist half turns) or (R U R' U')3 or (R U' R' U)3 (11 moves not counting AUF).
Faster alternatives:
(U) R U2 (R' M') U' M
(U) r U2 R' U' M
The concept isn't terrible per se, but you do need better "CPE" algs. If you keep the unsolved cross edge at DF or DB, your CPE is basically a subset of EODFDB as used in ZBRoux, and it can be done fully MU 2-gen.
(In fact, the only viable variant of this
is ZBRoux. Otherwise, you're just creating an additional step to fix the cross instead of just solving the cross properly in the first place. On very rare occasion (<1%) it might make sense to not immediately solve a full cross, but short of just using Roux or related methods, this is not something you should treat as a main method.)
May be of interest:
DF-DB swap + ELL (by Meep).