Q: Do most people in the UK think of members of the House of Lords as being MPs?
A: "Member of Parliament" is the formal designation of a member of the House of Commons. For example my local MP is, in full: Rt Hon Jeffrey M. Donaldson, Member of Parliament for Lagan Valley.
The title "Member of Parliament" or "MP" applies only to members of House of Commons. The phrase "Members of Parliament" is the plural form for members of the House of Commons.
To refer to MPs and members of the House of Lords as a group it is necessary to use a phrase such as "members of both Houses (of Parliament)" or "members of the Lords and Commons".
It is possible to use "members of parliament" to refer to Lords and MPs as a whole, but there is a serious risk of misunderstanding.
At the annual formal opening of parliament, at which MPs and Lords crowd into the Lords' chamber, the Queen addresses those present as "My Lords and Members of the House of Commons".
(The equivalent in Canada is the Speech from the Throne by the Governor General in which she addresses the members as "Honourable Senators, Members of the House of Commons".)