In his commentary on Leviticus 2:2, R' Samson Raphael Hirsch notes, as you do, the strong association in the Torah between the term אזכרה and the Mincha offering of flour, oil, and spices. In fact, he says that the concept of אזכרה is the whole purpose of the Mincha offering.
Note that the term אזכרה refers, in this verse, specifically to the handful of the offering that is to be burnt on the Altar. As the אזכרה, the handful stands in as the representative of the whole offering. From the way the verse is phrased, R' Hirsch infers (in a way I don't quite follow) that the whole offering is in the same way an אזכרה - a small subset that symbolically stands in for something larger. In particular, the flour, oil, and spice of the offering represent all of the material resources we need and have for, respectively, survival, comfort, and satisfaction.
The אזכרה function this offering performs is to get God to "remember" or apply "special care and attention" to that which it represents. That is, by offering the Mincha, we are praying to God to pay special attention to our physical needs. In so doing, we are acknowledging that for everything we need on every level, we are utterly dependent on God.
Based on all this, I think we can understand אזכרה as something like "a representative portion of a whole, which is used to pray for special attention to be payed to the whole." The remembrance/attention aspect is also present in the term זכרון. (Indeed, in this comment, R' Hirsch calls the אזכרה a "זכרון לפני ה".) However, I think the aspect of a part that stands in for the whole for this memorial purpose is unique to אזכרה.