Is the ionic solution of a half-cell part of the electrode in a electrochemical cell?
Your question is about definitions. 

I haven't read the Collins book, but, in general, the electrode is the metallic conductor that connects a circuit to the "non-metallic" part of the circuit. The electrolyte is this non-metallic part - dissociated ions that conduct electricity. Note electrolyte has broader uses than electrochemistry. 

The term electrode can be used in reference to the whole half-cell since it's the site at which the reaction is taking place. At other parts, all we have is storage of reactant and product for the reversible reaction. 
Consider the Standard Hydrogen Electrode(SHE), the whole half cell is referred to as an "electrode", though this is more convention than technical.

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I suspect the Collins book is for UK A-levels; they should use the clear general definitions;
the electrode is the solid metal connecting circuit to non-metallic part, &amp; the electrolyte is the solution of ions:

Here are some additional brief notes on contextual half cell setups:


SolidMetal/MetalSolution half-cell: Metal Electrode in metal ion solution.
All AqueousIons half-cell: Platinum electrode in solution.
Gaseous/AqueousIons half-cell: Glass/Platinum Deliver tube (for gas) bubble into solution.


The salt bridge is an electrolyte; A-level questions on electrode, electrolyte and salt bridge definitions aren't asked too often, consult your spec &amp; exam papers. Importantly, solutions need to be $1dm^3$,  at $298K$ and $100kPa$, &amp; salt bridge is typically $KNO_3$