CupofTea is in Ohio, USA, and as she said, she lives near Cleveland, up on or near Lake Erie.   The name of town in Ohio spelled Lima is pronounce LIE-ma.

In the UK, all lima beans may perhaps be referred to as butter beans. Not sure. In the USA, only some varieties of lima beans are called butter beans, or possibly broad beans. In the southern USA, the big, tougher varieties that are sold as dry beans as well as canned beans, and occasionally frozen, are called butter beans. They are a pale greenish tan to yellowish tan in color and taste very different from what we call lima beans, which are small, tender, green and not as starchy in taste and texture. In the southern USA, what we call lima beans are never sold dry, may be available fresh, but more likely available frozen, occasionally canned.

In the USA, the differences in regional accents result in some variation, but the differences in accents have to do almost entirely with regional differences in how vowels are sounded, or whether or not consonants get dropped or softened at the end of a word. (goin' vs going, for example.) A long vowel spoken by a person from New England will sound different from a long vowel spoken by a southerner, but both are recognized as the long vowel sound for that region, as another example. Regardless of what a dictionary may say, in the USA, the local regional pronunciation regarding the accented syllable, and whether it is a short or long vowel, is generally considered the correct pronunciation of the place name, regardless of what the dictionary says.

So, regardless of where one is from, to pronounce Lima, Ohio the same as Lima, Peru, would be a pronunciation, albeit understandable and forgivable the first few times one pronounced it incorrectly after being corrected. Ditto places like the place I mentioned upstream, Canaan Valley, WV. To pronounce it the same way one pronounces the biblical referenced place after which it is named, is an incorrect pronunciation. The USA is full of such place names.

There is a quite populated unincorporated area near me named Bahama. When I moved here many years ago, I pronounced it the same as I pronounce the Bahamas (as in the southern Atlantic islands.) Wrong, wrong, wrong. Bahama, NC is pronounced ba-HAY ma. Founded circa 1750, it's name was made from three prominent families who settled there, taking the first two letters of each of their surnames. (Ba)ll, (HA)rris, and (Ma)ngum.