Surprisingly difficult to pin down. Private label or Own brand seems to be the cover-all term but the difference between the one with the retailer's name and a different name is elusive. Perhaps some of these links will help you decide!

In Europe private labels, those generated by the retailer have now become a dominant issue. On average 45% of products sold in Europe are sold via private label, this compares with 25% in the USA.
But, what is a private label. It is a label unique to a specific retailer. These private labels can be divided into a number of groups.
Store brands
This is where the retailers name is very evident on the packaging
Store Sub-brands
Products where the retailers name is low-key on the packaging.
Umbrella branding
A generic brand independent from the name of the retailers name. Umbrella brands are used in different product categories.
Individual brands
Name used in one category, this is only used to promote a ‘real’ discount product line.
Exclusive brands
Again a name used in one category, but to promote ‘added value’ products within the category.
http://209.85.135.104/search?q=cache:QOQufgnGUtUJ:retailindu...
Own-label sales generate 38 per cent of Sainsbury's total revenue, with its Taste The Difference premium range estimated to contribute between £200m and £300m.
Supermarkets have moved on from creating imitation brands to developing own-label sub-brands that stand as brands in their own right. These brands have become an everyday feature of consumers' lives and, as a result, have earned loyalty.
In the food market, the one category that own-label sub-brands dominate is chilled foods. In 2001, own label accounted for 98 per cent of the UK chilled foods market (source: Mintel).

But the food sector is becoming saturated, so that many supermarkets have explored non-food markets as an area in which they can develop own labels. Tesco's plan to extend its Finest range into homeware is part of that shift in direction.

Tesco's other sub-brands include Healthy Eating, Organic and Kids, but Finest is its most successful, growing rapidly at a rate of 30 per cent a year. Next year, Tesco expects to generate £600m in sales from the Finest sub-brand alone.
One of the most famous own-label brands to have emerged from the UK supermarket chains is Asda’s “George” clothing range. This is estimated to be worth around £1bn in sales – making Asda one of the leading clothing retailers in the UK.
http://209.85.135.104/search?q=cache:ptoNzditX2QJ:www.tutor2...
Store brands (or own brands in the UK) are brands which are specific to a retail store or store chain. The retailer can manufacture goods under its own label, re-brand private label goods, or outsource manufacture of store brand items to multiple third parties - often the same manufacturers that produce brand label goods.
http://en.wikipedia.org/wiki/List_of_store_brands
With the emergence of strong retailers, the "own brand", the retailer's own branded product (or service), emerged as a major factor in the marketplace. Where the retailer has a particularly strong identity, such as, in the UK, Marks & Spencer in clothing, this "own brand" may be able to compete against even the strongest brand leaders, and may dominate those markets which are not otherwise strongly branded. There was a fear that such "own brands" might displace all other brands (as they have done in Marks & Spencer outlets), but the evidence is that—at least in supermarkets and department stores—consumers generally expect to see on display something over 50 per cent (and preferably over 60 per cent) of brands other than those of the retailer. Indeed, even the strongest own brands in the United Kingdom rarely achieve better than third place in the overall market. In the US, Target has "own" brands of "Market Pantry" and "Archer Farms" each with unique packaging and placement.
http://en.wikipedia.org/wiki/Brand#Own_.28Private_Label.29_b...
Own-label products have always existed, and have been the cornerstone of Marks & Spencer’s success story over the last 100 years.

But among other retailers they were widely regarded as cheaper, inferior versions of branded products.

This image was confirmed in the Eighties, when the major supermarkets brought out "value label" products - cereals, for example, came in white plastic bags with muesli stamped in large black letters - which were heavily discounted.

However, in the last decade, supermarket own-label brands have gradually become more sophisticated.

Rather than operating a single own-label brand, Britain’s big four supermarkets - Tesco, Asda, Sainsbury, and now Safeway/Morrisons - have brought out a number of sub-brands that cater for different customers.

All still operate a limited value range for the price- sensitive shopper, a regular brand which sits in the middle ground and a premium brand, such as Asda Extra Special.
Beyond this there are sub-brands aimed at organic shoppers, children (Sainsbury’s Blue Parrot) and healthy eating (Tesco Healthy Living).

The result, said Prof Hughes, is that the UK is the world leader in house brands. "We are probably more sophisticated at this than any other country," he said.
http://209.85.135.104/search?q=cache:ABgPQAAuhboJ:thescotsma...
With rare exceptions, they do not carry a retailer's name in the way own label brands of other products usually do.
http://www.bmj.com/cgi/content/full/316/7135/929