I think the term "lexical aspect," imprecise as it is, will be understood as "non-grammatical aspect" by exegetes. So we're not really talking about lack of clarity. But it does beg questions about what's going on, and that may be inappropriate. You see, different theorists have somewhat different ideas about how the aspect of these sentences are built up, and they disagree about how much is handled by the lexicon and how much is handled by the syntax and how much by the context, and there are different ways of slicing up the division of labor. For example,Jonathan Robie wrote:OK, but let me try to understand your objection to the term "lexical aspect".
Most aspects of meaning of a verb are affected by the objects of the verb. Sure, "John walked in the park" and "John walked to the park" have different lexical aspects, but "John hit a ball in the park" and "John hit on her in the park" also have different meanings. I generally think of the various meanings of a verb as being lexical, even if a single verb can have many senses, which may sometimes be selected based on the objects of the verb.
Is my thought model flawed here? If not, then "lexical aspect" still seems like a useful term, it's just that the aspect for a given lexeme may need to be parameterized further.
1. The approach you've described is a polysemy approach, where the lexicon for walk would have one entry that's telic and another that's atelic. Presumably context (here the prepositional phrases) would disambiguate which sense of walk is meant.
2. Another approach is a coercion approach, where the lexicon for walk would say that its lexical aspect is atelic (activity), but the PP to the park causes the lexical aspect of walk to be coerced into a telic accomplishment. (Conversely, one could claim that walk is telic and the PP in the park coerces the predication to atelicity.)
3. Yet another approach is a compositional approach, where (non-grammatical) aspect can be assigned to verbs, participants, and adjuncts. In this approach, walk may not have a "lexical" aspect at all, but that the non-grammatical aspect comes from the PP phrases, either the atelic in the park or the telic to the park. This sort of analysis is more popular among those working in Finno-Ugric languages where the aspect of the predication is encoded by the case of the object (genitive for perfective, and partitive for imperfective).
I'm sure there are other combinations of lexical, syntax, and grammar too.
So depending on which approach you take, the work of aspect can be handled in the lexical, the syntax, or context, or in various combinations of them. And linguists are still fighting over which approach can give the best explanatory value, best fit, and/or smallest grammar. The term lexical aspect picks a particular side in this debate almost by definition. Ideally, one would want a term less prejudicial.