I have also gotten conflicting advice, similar to what you have heard. I'm no expert, but I'll tell you what I ended up doing which seemed to work out pretty well, at least so far.

We bought a ~100 year old home six months ago that we're just getting around to painting now, after rewiring. Most of the trim is painted with what seems to be a high gloss oil-based paint, and a much of it (especially doors and baseboards) is quite dirty. The only trim I've painted over so far is a few doors, but I'll be starting on the rest of the trim in a couple weeks.

For the doors, I scraped off just whatever bits were already flaking off, then washed them really well, and then used "liquid sandpaper". If your trim is glossy you may want to use liquid sandpaper to remove the gloss. It's not a paint stripper, so you won't be left with messy globs of dissolved paint. It just removes some of the sheen so that the new layers bond to it better.

Then I primed with an oil-based primer. I have used both latex and oil primers in a previous home, and the oil primers always seem to bond better.

Then painted with Regal Select Semi Gloss. I haven't tried Natura, but I might...

Anyway, I would recommend liquid sandpaper rather than sanding as long as the old paint is basically well-bonded to the trim, then oil-based primer, than latex paint.

I'm curious what suggestions other people have..

All the trim needs to be cleaned first., lightly sanded, cleaned. Then you can apply any QUALITY, BONDING primer( oil or latex), I would use latex ( Zinsser's 123), then apply the 2 top coats, again quality paint( meaning a trip to a real paint store).
As to the "painters" #1 ok but you need to sand first(lightly)
#2 forget about him
#3 ok, to a point,, you do not HAVE to use oil based primer, you can and it will work just fine

Thanks to you both!!

I wouldn't choose any of the 3 painters actually. If you are saying you MUST use oil or you MUSt use latex, then you are clearly not keeping yourself educated. The key as always, is the prep. Also, I have never used liquid sandpaper and I'm sure it works to dull the sheen, but I think sanding the old fashioned way is the best way to prepare trim for paint. Some people just hate sanding though, which is why liquid sandpaper is around and also why manufacturers are trying to make primers now that will stick without sanding. If you are hiring a pro though, they should be sanding, the end.

If you are hiring a pro though, they should be sanding, the end.

sorta what I said