I yield.

Every kokugo: 〔接続助詞〕
DOJG: "A conjunction which indicates that the preceding clause expresses a conditional"
JMdict's own entry for 接続助詞: "conjunction particle (e.g. "ba", "kara", "keredo", "nagara")"

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I very much agree with what non has said previously, ば is both a particle and a conjunction. It would be inconsistent both with other entries like けど, けれど and with every major reference, to dismiss this. I think non has sufficiently explained why this should have [conj] tag (apart from sense three)

@@ -9,0 +10 @@
+<pos>&conj;</pos>
@@ -14,0 +16 @@
+<pos>&conj;</pos>
@@ -24,0 +27 @@
+<pos>&conj;</pos>
@@ -29,0 +33 @@
+<pos>&conj;</pos>

@@ -21 +21 @@
-<gloss>why don't you...?</gloss>
+<gloss>why don't you ...?</gloss>

I did not think any references would be needed, It is formally classified as a 接続助詞 just about everywhere: daijs, daijr, smk, meikyo, koj...

ば's primordial function is to connect two clauses, clause A to clause B, where A is in some manner precedent to B, this should be in line with the citation you have just provided of a conjunction. 

Additionally, I do not see how these definitions would - or even could - be mutually exclusive: every single conjunction has a grammatical aspect to it, it must always form a syntactically subordinating or coordinating relation in between the two linked items; dismissing [3] as an omission of clause B, and with the potential exclusion of [2], ば falls into the former type.
I must also bring to attention that the definition of a conjunction provided is vague and potentially problematic, for example: how does it deal with a coordinating conjunction between two nouns? ”僕と君と...”or anything following the same structure will be constituted of two words that are not particularly large units of meaning, certainly not clauses, and I do not think most would consider each a phrase. Would と cease to be a conjunction? would also または or および? This is not limited to Japanese either, you can just as easily take "Me and you..." and the same problem arises.

Lastly, not classifying ば as a conjunction makes the corpus inconsistent: other terms formally classified as 接続助詞 such as と, けれども and its variants have both [conj] and [prt] on their entries; notably, なら and たら both directly derive from ば and are classed as [conj] (though the former's entry is itself a tad inconsistent).
Now, as the meaning does not change even in their original forms of ならば and たらば, if ば is not a conjunction, what is it? or rather, if it is not, how are they?