Is a solicitor a lawyer?
Yes — and so are several other kinds of person, which is the part the question is really getting at.
"Lawyer" is not a protected title in England and Wales and describes no single qualification. "Solicitor" is protected, and so is "chartered legal executive". Both are regulated professionals who can be instructed directly by a client, and both appear on a public register maintained by a regulator — solicitors on the SRA's, chartered legal executives on CILEx Regulation's.
What differs is scope. A solicitor qualifies broadly and may then specialise. A chartered legal executive qualifies in a specialism and adds rights to it one at a time — there are 30 distinct practice rights on this register, and 1,685 people hold at least one that entitles them to conduct litigation or appear as an advocate. The question to ask about anyone is not which title they hold but what they are authorised to do, which is the thing the registers actually record.