I am about to be a little pedantic, I don’t mean anything mean by it but it is just to make sure we are speaking about the same things.
By “some kind of cartridge system” I assume you mean a cartridge/converter system where you can either use a pre-packaged cartridge, or else a converter that allows you to use bottled ink, yes? You are correct that the cartridge or the converter can become detached inside the pen and then leak, although I have never had a problem with that.
And by “pump-style reservoir” I’m going to make the leap and surmise that you mean a piston-filler, where you turn a knob on the end of the pen barrel (with the nib submerged in the ink) that moves a piston forward towards the nib, and then by turning in the other direction moves the piston away from the nib and draws ink in by suction. That kind of pen stores all the ink (usually) directly inside the barrel.
There are some other variations on filling systems, and any fountain pen can leak, but a well-made piston filler pen is usually very reliable. The only drawback I have found is that they can be tedious to clean out, which you should do maybe a couple of times a year even if you are always using the same ink.
The Pilot Metropolitan is a variant of the converter system, and I suspect is not what you want. So is the Lamy Safari and all its variants.
For a reliable and inexpensive piston-filler pen, I recommend pens from TWSBI, a company based in Taiwan. You can order them with a variety of nib widths (such as fine, medium, broad, stub), and you can also buy replacement nibs if the one you picked out doesn’t suit you. They are available at a variety of online dealers such as gouletpens.com and jetpens.com (I have no affiliation with them, but I have bought from them both myself). I would avoid buying pens from Amazon or eBay unless you really know what you are doing; you might save a few dollars, but the risk is great, and the after-purchase service is probably nil.
Good luck, I’m glad to see someone looking at fountain pens for regular use.