@nm I see that for you Physics:-Version() reports the location of Physics in the Maple 2018 lib folder.
That should just mean that the updates are not installed. They ought to be in the equivalent place reported here for my updates:
"C:\Users\Bruger\maple\toolbox\2018\Physics Updates\lib\Physics Updates.maple", `2018, December 21, 13:9 hours, version in the MapleCloud: 264, version installed in this computer: 264`
where 'Bruger' is my user name.
Notice that Tom Leslie reports the equivalent location of his Physics Updates.
That the message you get reports "... version installed in this computer: 264"  is puzzling, but may just be one thing that "worked" during your attempts at installation. But why does it say: "... version in the Maple Cloud: unable to determine" ?
Surely you know all this, so sorry: not much help.
 

@nm I just downloaded the Physics update from the Cloud and in Maple 2018.2 64bit on a computer with Windows 10 and last updated a day ago.
I experienced no problem during download. The pde problems posted by Edgardo Cheb-Terrab recently on MaplePrimes were solved with no errors.

@ecterrab Yes I didn't mention it, but did indeed try it. What I hoped for was that Maple picked the correct branch in each case itself.
Actually, I only needed splitting at x=Pi/2. But as it turned out that would have hidden the use of the wrong branch on Pi/2..Pi since the two integrals would be zero both of them regardless of branch.

@Rouben Rostamian  You are right. Finding the inverse of cos restricted to an interval different from 0..Pi seems not to be implemented in Change, which uses solve.
solve gives the same result for all four:
 

solve({cos(2*x)=z,x>0,x<Pi/4},x);
solve({cos(2*x)=z,x>Pi/4,x<Pi/2},x);
solve({cos(2*x)=z,x>Pi/2,x<3*Pi/4},x);
solve({cos(2*x)=z,x>3*Pi/4,x<Pi},x);

It has been a long time since I used Mathematica, but I remember it issuing a warning about having to invert transcendental functions.

@Mariusz Iwaniuk I'm not convinced. The method used is ftoc (try infolevel[int]:=5), but the antiderivative is discontinuous.
No indication that that is taken into account.

u:=convert(cos(2*x)/(1+2*sin(3*x)^2), exp);
U:=int(u,x);
plot(Re(U),x=-1..Pi+1);

@vv If we use the integral in the form int(-cos(2*x)/(-2+cos(6*x)), x ) instead, it appears by plotting (!) that that antiderivative is continuous. As pointed out in my answer Maple doesn't find the definite integral with the integrand written like that though.

@Carl Love Thanks, Carl, for your detailed explanation. I think I got it now. I'm slow.

@ecterrab The other day I deleted the Physics Update folder (named 2018) and after that the warning disappeared.
The point of that exercise was to find out if the warning would go away. Since it did, the warning will only come up for people actually downloading the Physics update. Thus they would presumably know that they have such a thing.
Thus I'm not too worried anymore.

Clearly, the hack above can be turned into a procedure.
Input the dsolve solution (res) and the number of odes of the corresponding first order system (ndiff).

restart;
FreezeParams:=proc(res,ndiff::posint,$) local R,params,ind,i,j,S;
  params:=rhs~(res('parameters'));
  ind:=[ListTools:-SearchAll('undefined',params)];
  R:=convert(eval(res),string);
  S:=NULL;
  j:=0;
  for i from 1 to nops(params) do
    if not member(i,ind) then 
      j:=j+1;
      S:=S,cat("Y",[ndiff+j])=convert(params[i],string)
    end if;
  end do;
  R:=StringTools:-Subs({S},R);
  parse(R)
end proc:
###  
## Example.
res:= dsolve({diff(y(x),x,x)=a*y(x)+b, y(0)=c,D(y)(0)=d}, numeric, parameters= [a,b,c,d]):
res0:=FreezeParams(res,2); 
res(parameters);
res0(parameters);
res(parameters=[a=1]);
res1:=FreezeParams(res,2); 
res0(parameters);
res1(parameters);
res(parameters=[b=2,c=3]); # Two more set
res2:=FreezeParams(res,2);
res2(parameters);
res2(parameters=[7,8,9,10]); #Attempting to set parameters
res2(parameters); # Not set
res0(parameters); #Unchanged
res1(parameters); # Unchanged
res(parameters=[d=4]); # All set.
res3:=FreezeParams(res,2);
result:=res(1); 
res3(1);
res(parameters=[0,0,0,0]);
res(1);
res3(1); # same as 'result'.

@Carl Love It seems that the solution procedure in dsolve/numeric uses Y[2] for the parameter in the case of one parameter in a system of one equation:
 

restart;

res:= dsolve({diff(y(x),x)=a*y(x), y(0)=1}, numeric, parameters= [a]):
res(parameters=[2]);
result:=res(1);
R:=convert(eval(res),string):
R2:=StringTools:-Subs("Y[2]"="2.",R):
res2:=parse(R2):
res2(1); # As result above
res2(parameters);  # Query
res2(parameters=[7]); # Seemingly set
res2(1); # But it isn't

Guessing the (obvious) generalisation:

restart;
### First order, two parameters:
res:= dsolve({diff(y(x),x)=a*y(x)+b, y(0)=1}, numeric, parameters= [a,b]):
res(parameters=[1,2]);
result:=res(1);
R:=convert(eval(res),string):
R2:=StringTools:-Subs({"Y[2]"="1.","Y[3]"="2."},R):
res2:=parse(R2):
res2(1);
res2(parameters); 
res2(parameters=[7,9]);
res2(1);
######################################
restart;
### Second order, two parameters:
res:= dsolve({diff(y(x),x,x)=a*y(x)+b, y(0)=1,D(y)(0)=0}, numeric, parameters= [a,b]):
res(parameters=[1,2]);
result:=res(1);
R:=convert(eval(res),string):
R2:=StringTools:-Subs({"Y[3]"="1.","Y[4]"="2."},R):
res2:=parse(R2):
res2(1);
res2(parameters); 
res2(parameters=[7,9]);
res2(1);

@Carl Love Maybe I'm misunderstanding your intention, but the localized version still knows about the parameter 'a' and that can still be set:
 

restart;
Sol:= dsolve({diff(y(x),x)=a*y(x), y(0)=1}, numeric, parameters= [a]):
#
#"Carl's Magic .m" method:
Localize:= proc(Sol, params::list(complexcons))
option remember;
   Sol('parameters'= params);
   sscanf(sprintf("%m", eval(Sol)), "%m")[]:
end proc:
##
res:=Localize(Sol,[1]);
res(parameters);  #        [a = 1.]
res(1); #                  [x = 1., y(x) = 2.71828133411964]
res(parameters=[2]); #     [a = 2.]
res(1); #                  [x = 1., y(x) = 7.38905384345779]          

I thought that you wanted a procedure like the one you get when using dsolve/numeric for a concrete value of 'a', here a = 1.
In fact Sol and res above are identical:

res:=Localize(Sol,[1]);
interface(verboseproc=3);
S:=convert(eval(Sol),string):
R:=convert(eval(res),string):
StringTools:-Compare(S,R); # true
StringTools:-Compare(R,S); # true

The procedure(s) are terrifyingly long.
I checked that R (and so S) actually has the information about the parameter value by setting the parameter to 7.23456789 instead of 1.
Then I did

StringTools:-Search( "7.23456789", R );
R[1279..1306];   #  "Array(1..2, [1.,7.23456789])"

###
Furthermore, if you use the "naive" approach, but add option remember it seems that you get good results for the plot3d problem:
 

restart;

Sol:= dsolve({diff(y(x),x)=a*y(x), y(0)=1}, numeric, parameters= [a]):

Q:= proc(Sol, params::list(complexcons))
option remember;
   Sol('parameters'= params);
   Sol
end proc:

Y:= (A,X)-> 
   `if`([A,X]::[numeric$2], eval(:-y(:-x), Q(Sol, [A])(X)), 'procname'(args)):
gc():
P2:= CodeTools:-Usage(plot3d(Y(A,X), A= -2..2, X= -2..2, grid= [99$2])):

@Carl Love You can save the procedure as an .m file.
 

restart;
ode:=diff(x(t),t)=x(t)+a;
res:=dsolve({ode,x(0)=1},numeric,parameters=[a],output=listprocedure);
X:=subs(res,x(t));
res(parameters=[5]);
save res, "F:/MapleDiverse/temp.m";
seq(X(i),i=1..10);
res(parameters=[7]);
seq(X(i),i=1..10); 
read "F:/MapleDiverse/temp.m";
X:=subs(res,x(t));
seq(X(i),i=1..10); 

I should have added these lines at the bottom:
 

res(parameters=[7]);
seq(X(i),i=1..10);

Is your statement:

"It turns out that setting the parameter's numeric value in the
dsolve numeric solution-procedure causes the loss of previous details
of the numeric solving, even if the parameter's value is the same."

documented somewhere?
Or how to see that it is actually happening?

@siddikmaple Try this:
 

CSTR:={p[b], p[s], p[0, b], p[0, c], p[0, p], p[0, s], p[1, b], p[1, c], p[1, p], p[1, s], p[2, b], p[2, c], p[2, p], p[2, s], p[3, b], p[3, c], p[3, p], p[3, s], r[0], r[1], r[2], r[3], t[0], t[1], t[2], t[3]}=~(0..1);
ans := DirectSearch:-SolveEquations([e1, e2, e3, e4, e5, e6, e7, e8, e9, e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, e20, e21, e22, e23, e24, e25, e26],CSTR, evaluationlimit = 1000000);

The answer doesn't seem to satisfy the equations very well though. Is there a solution at all?
 

If fsolve returns unevaluted then it wasn't able to find a solution. It could help to give it a starting point or ranges . See the help for fsolve.
To get better help upload a worksheet with the equations. Use the green arrow in the MaplePrimes editor to do that.