The following procedure $p$ has its arguments fully evaluated at the top level, but evaluated to only one level when called within the (anonymous) procedure shown.
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p := proc( x ) print( x ); end proc;

${p}{:=}{\mathbf{proc}}\left({x}\right)\phantom{\rule[0.0ex]{0.5em}{0.0ex}}{\mathrm{print}}{}\left({x}\right)\phantom{\rule[0.0ex]{0.5em}{0.0ex}}{\mathbf{end\; proc}}$
 (1) 
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proc()
local a, b;
a := b;
b := 2;
p(a);
end proc();

Modifying the formal parameter $x$ of $p$ changes it so that it has special evaluation rules.
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p := proc( x::uneval ) print( x ); end proc;

${p}{:=}{\mathbf{proc}}\left({x}{::}{\mathrm{uneval}}\right)\phantom{\rule[0.0ex]{0.5em}{0.0ex}}{\mathrm{print}}{}\left({x}\right)\phantom{\rule[0.0ex]{0.5em}{0.0ex}}{\mathbf{end\; proc}}$
 (6) 
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proc()
local a, b;
a := b;
b := 2;
p(a);
end proc();

The procedure evalf has special evaluation rules because it ensures that its second argument (if provided) is evaluated before its first argument.
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evalf( proc() print( FIRST ); Pi end(), proc() print( SECOND ); 15 end() );

The procedure assigned uses a nonstandard evaluator (evaln) when it evaluates its argument.
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$\mathrm{assigned}\left(\mathrm{H2}\right)$

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$\mathrm{evaln}\left(\mathrm{cat}\left('H'\,2\right)\right)$

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$\mathrm{assigned}\left(\mathrm{cat}\left('H'\,2\right)\right)$

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$\mathrm{evaln}\left(H\Vert 2\right)$

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$\mathrm{assigned}\left(H\Vert 2\right)$

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$\mathrm{H2}\u2254\mathrm{sin}\:$

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$\mathrm{evaln}\left(H\Vert 2\right)$

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$\mathrm{assigned}\left(H\Vert 2\right)$

You can control the order of argument evaluation by suppressing it entirely and making the evaluations explicit in the desired order.
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p := proc( a::uneval, b::uneval )
local ea, eb;
eb := eval( b ); # evaluate second
ea := eval( a ); # evaluate first
[ ea, eb ]
end proc:

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p( proc() print(FIRST); 2 end(), proc() print(SECOND); 3 end() );

$\left[{2}{\,}{3}\right]$
 (17) 