In general, legislatures do not require the right of initiative (or even the right of amendment) to be called a legislature or for their members to be called legislators.
A specific example is the Dutch Senate, members of which don't have either right (initiative or amendment), yet it is called a legislature and it is the upper chamber of the States General of the Netherlands which is a bicameral legislature.
In particular, I expect to see similar situations in other bicameral systems (especially when not modelled after the Westminster system which does have right of initiative and amendment in both chambers).
In the case of MEPs
Do MEPs still meet the definition of a legislator?
Yes, because the European Parliament is a legislative body. The definition of legislative body from Wiktionary:
A political institution which holds the legislative power in a state, and often controls the executive power.
The Parliament is one of the bodies involved in the ordinary legislative procedure, according to Wikipedia:
Article 294 TFEU outlines ordinary legislative procedure in the following manner. The Commission submits a legislative proposal to the Parliament and Council.
At the first reading Parliament adopts its position. If the Council approves the Parliament's wording then the act is adopted. If not, it shall adopt its own position and pass it back to Parliament with explanations. The Commission also informs Parliament of its position on the matter. At the second reading, the act is adopted if Parliament approves the Council's text or fails to take a decision. The Parliament may reject the Council's text, leading to a failure of the law, or modify it and pass it back to the Council. The Commission gives its opinion once more. Where the Commission has rejected amendments in its opinion, the Council must act unanimously rather than by majority.
In practice, this gives the Parliament and the Council somewhat equal importance, according to Wikipedia:
With each new treaty, the powers of the Parliament, in terms of its role in the Union's legislative procedures, have expanded. The procedure which has slowly become dominant is the "ordinary legislative procedure" (previously named "codecision procedure"), which provides an equal footing between Parliament and Council.
In particular, under the procedure, the Commission presents a proposal to Parliament and the Council which can only become law if both agree on a text, which they do (or not) through successive readings up to a maximum of three. In its first reading, Parliament may send amendments to the Council which can either adopt the text with those amendments or send back a "common position". That position may either be approved by Parliament, or it may reject the text by an absolute majority, causing it to fail, or it may adopt further amendments, also by an absolute majority. If the Council does not approve these, then a "Conciliation Committee" is formed.