I will first state the dictionary entry, and then other formations that also
yield the dictionary entry.

Consider the following :

  1. 思える - to seem/appear, 

    * potential form of 思う - can think
  2. 合わせる - to join together, 

    * potential form of 合わす - can join together

    * causative form of 合う - to let suit/match/agree with

  3. 解ける - to come untied / be solved

    * potential form of 解く - can untie/solve
  4. 空ける - to empty

    * potential form of 空く - can become empty

As above, there are words that in their potential/causative form that already
have a dictionary meaning that is different from their conjugated forms.

**(Question)** Were they equal in meaning at some point in time? Or are they
completely different words and should be treated as such?
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2011-10-11T01:40:52.473
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2011-10-11T05:30:48.243
2011-10-11T04:16:07.620
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question
grammar meaning potential-form causation
Dictionary entry and derived forms, were they equal at some point?
Not all of what you claim as potential form are potential forms. Actually, all
`-e-` except for the one in `思える` just switch transitivity.

You seem to be doing the conjugation wrong. Notice that when the verb stem
ends with a vowel, the potential affix is `-re-` instead of `-e`, which is
only for when the verb stem ends with a consonant.

> omow-u → omou (tr.) 'think'  
>  omow- **e** -ru → omoeru (tr. potential) 'can think'
>
> aw-u → au (it.) 'meet'  
>  aw- **e** -ru → aeru (it. potential) 'can meet'
>
> aw-ase-ru (tr.) 'meet'  
>  aw-ase- **re** -ru (tr. potential) 'can meet'
>
> tok-u (tr.) 'solve'  
>  tok- **e** -ru (tr. potential) 'can solve'
>
> tok-e-ru (it.) 'dissolve'  
>  tok-e- **re** -ru (it. potential) 'can dissolve'
>
> ak-u (it.) 'open'  
>  × ak- **e** -ru (it.) 'can open'
>
> ak-e-ru (tr.) 'open'  
>  ak-e- **re** -ru (tr. potential) 'can open'
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2011-10-11T02:11:31.630
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2020-06-17T08:18:27.500
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answer