Opinions will vary, every runner can only really comment from their own experience. If I was starting out running again I would get some mid-range (price wise) neutral road shoes. That way you wont lose much if running is not for you and you'll still have a comfy pair of trainers
By the time the weather deteriorates you'll probably be looking for a second pair anyway, then you'll have a better idea of your running style: neutral, over pronator, or supinator from the pattern of tread wear (not a perfect gauge but the
wet foot test has it's critics too). Your running style will still be developing anyway.
Maybe I've just been lucky but I've stuck with neutral shoes and they've been fine. Most runners I know have done this until they suspect there might be a problem (very few and even then some shoes are fine whilst others are not - even of the same "type", it's a personal thing, and trial and error). Some have gone down the assessment route, in extremes even leading to orthotics, which I've heard both good and bad about.
I've had track spikes, cross country spikes, road shoes, racing flats (road) and fell studs. Personally I've never felt the need to have trail shoes and I don't think they would be suitable for road use other than short stretches. On the other hand I've raced successfully on trails with road racing flats (in Summer).
A lot of the jargon and hype is all about selling shoes. There is growing opinion that shoe development has gone too far and can in some cases lead to injury, hence the return to barefoot and natural styles. The worst shoes I ever had were also the most expensive. Advice from a proper running shop is a good call (
@MikeG) as is the comment above about excessive heel cushioning when off-road (
@ayceejay ).
Running should be kept as simple as possible, don't over-think it, just get some comfy shoes and enjoy it!