excellent and (prob unintentionally deep) question. there are indeed halting-detecting programs that can succeed on limited sets of inputs. its an active area of research. it has very strong ties to (automated) theorem proving areas.
however computer science does not appear to have an exact term for "programs" that "sometimes" succeed. the word "algorithm" is usually reserved for programs that always halt.
the concept seems to be distinctly different than probabilistic algorithms where CS theorists insist there be some known or computable probability on their succeeding.
there is a term semialgorithms that is used sometimes but its apparently a synonym for recursively enumerable or noncomputable.
so for purposes here, call them quasialgorithms. the concept is different than decidable vs undecidable.
one might say that one cannot compare quasialgorithms. but in fact there seems to be a natural hierarchy (a partial ordering) of these quasialgorithms. suppose a quasialgorithm $A$ can detect halting of some limited set of input programs say $X$. another one $B$ can detect halting of a set $Y$. if $X \subset Y$ ie $X$ is proper subset of $Y$ then $B$ is "more powerful" than $A$.
in CS this "quasi algorithm hierarchy" seems to be studied mostly only informally so far.
it shows up in busy beaver research and the PCP problem. in fact a DNA based computing attack on PCP can be seen as a quasialgorithm. and its seen in other areas already noted such as theorem proving.
 New millenium attack on the busy beaver problem
 Tackling Posts correspondence problem by Zhao (v2?)
 Using DNA to solve the Bounded Post
Correspondence Problem by Kari et al
 proving program termination by Cook et al, Comm. of the ACM
(so this is actually a very deep question that defn deserves to be on TCS.SE imho... maybe someone can re-ask it there in a way such that it fits & stays)