:::::Objection #1: 
 :::::A 16-bit app on a register-oriented machine must, obviously, not use registers wider than the maximum register width. -)  However, that doesn't prevent it from having data items wider than the maximum register width.  Late-1970's versions of C on the 16-bit PDP-11 had a 32-bit integral data type . 
 :::::The {{n-bit}} template just says "In computer architecture, {{{1}}}-bit integers, memory addresses, or other data units are those that are at most {{{1}}} bits {{{2}}} wide."  That specifically refers to computer architecture, so it allows for wider integer arithmetic, which would have to be done with multiple arithmetic instructions. 
 :::::64-bit computing, which doesn't use that template, says "In computer architecture, 64-bit computing is the use of processors that have datapath widths, integer size, and memory addresses widths of 64 bits (eight octets)."  This is similar; it speaks of "processors", but it's still talking at the instruction set level. 
 :::::I'm not convinced that "In computer architecture, 16-bit computing is the use of primarily 16-bit data, including integers and memory addresses.  That is, data elements are at most 16 bits (2 octets) wide.", from Lyon's version, is an improvement.  It doesn't change it from something that doesn't allow, for example, a 32-bit  to something that does.  It might better handle the PDP-11, in which one of the floating-point instruction sets (the one introduced with the 11/45's floating point processor) provided limited support for 32-bit integers (converting to and from floating-point, but no arithmetic), or the 80286, with its 24-bit segmented addresses, but it does so by adding the somewhat weasel-wordish "primarily". 
 :::::Objection #3: 
 :::::"Popular" varies with time and with market segment.  16-bit minicomputers were quite popular in the computing market throughout the 1970's, and 16-bit personal computers were quite popular during at least the first half of the 1980's.  However, you won't find many 16-bit minicomputers or personal computers these days.  You may find lots of 16-bit microprocessors in embedded applications now.  I'd say that any talk of "popularity" belongs in text giving more detail than just "CPU and ALU architectures based on registers, address buses, and data buses of 16-bit width have been popular for decades.", giving some historical background.  That can, BTW, be done whilst still using the {{n-bit}} template - put it in another section, another paragraph, or another sentence in that paragraph, after the template. 
 :::::So, for now, I'd say "leave things as they are, pending further discussion".  Lyon's rationale for the rename of 64-bit to 64-bit computing, and some discussion of the issues with the {{n-bit}} template can be found at Talk:64-bit computing#Title.  More of his arguments against the template can be found at Template talk:N-bit#Longlasting problems from ancient template.  I would suggest that further discussion of the merits of the template be done there, and that replacing the template in this or other "n-bit" articles not be done until something approximating a consensus is reached there.  I'm not sure where the merits of renaming "n-bit" to "n-bit computing" should be discussed, but I'd like to see that discussed a bit more and, again, something approximating a consensus reached before renaming any other articles.