The short answer to this question is NO.
To understand the reason one needs to appreciate certain things about proteins and about research into them. A primary objective in research into proteins is to understand how their structures determine how they work, not to classify them by function. Of course, assigning similar proteins to groups and using phrases to describe them is helpful, but as science progresses broad categories may be found to be invalid or unhelpfully wide, and even when valid it may be more helpful to consider narrower categories, with the practitioners being aware of the overall picture.
I have given introductory lectures on proteins to undergraduate students, and would list some of the diverse roles they can play. I might have said something like “some catalyse reactions, some play a structural role, some transport substances across membranes, others carry molecules from one location to another…”. For stylistic reasons I have used two different verbs “transport” and “carry” — and if I wanted to make headings for a list I might use “carrier proteins” and “membrane transporters”, but these are not official categories and others might use other descriptions.
As @bryankrause commented, “carrier transport proteins” such as apolipoprotein, albumin, zinc carriers, haemoglobin etc represent very diverse types and are generally not considered together after first introduction. The term “carrier protein” has problems as it may be mentally associated with “acyl-carrier protein”.
As for the “membrane transporters” it is instructive to look at the index to the respected text, Berg et al. Biochemistry — Chapter 13 which considers such proteins under the title “Membrane Channels and Pumps”. This has a subheading “Many molecules require protein transporters to cross membranes” but then proceeds to use specific terms for different classes like “ion channel proteins”, “secondary transporters”, “ion pumps”.
In conclusion protein researchers tend to focus mainly on descriptive phrases and precise specialist sub-categories when talking about functional groups. The nearest thing to biological taxonomy is perhaps structural classifications such as SCOPe.