Can I travel on ESTA for personal reasons if I have a B1 visa?

There's a bit to unpack here...

Before getting to your question, I'd suggest checking your visa to see exactly what type it is. It's rare for the US to issue a "B1" visa to a national that is eligible for the Visa Waiver Program - more normally they would issue a "B1/B2" visa which can be used equally for business or tourism. This doesn't fundamentally change the answer below, but it's worth understanding.

Now, lets talk about "duration of status". ie, how long you can stay in the us on a single trip. If you enter the US under the Visa Waiver Program (ie, "using" your ESTA) you will be granted a duration of status UP TO 90 days. Normally this will be 90 days, but it can be less - we'll get to that shortly. If you enter using a Visa (B1 or B1/B2), then you will be granted a duration of status of UP TO 180 days. Again, normally 180, but can be less. This is where your type of visa becomes important, as if you are entering in B2 status (using the visa) you'll likely get a longer potential stay than entering under the VWP program (normally deemed 'WB2' status)

The number above control how long you can stay on a single trip. Once you leave the us (with a few exceptions such as a short trip to Canada/Mexico), your status for that trip is over, and the next time you enter you'll be given a new duration of status, for another "up to" 90/180 days.

OK, so lets talk about "up to". Border officials will normally give you the full 90 (VWP) or 180 (Visa) days - unless they have a reason not to. One of the reason they can give you less is if they believe that you are spending too much time in the US - in effect if you are attempting to "live" in the US.

There's no formal rule for what is considered "too much time", but the general (documented) rule of thumb is that you should be spending more time outside of the US than inside the US. ie, if you come to the US for ~3 months and then leave, then you would generally be expected to spend ~3 months out of the US before you were admitted again - especially if you were entering under the VWP. If you were to return after only (say) 1 month outside of the US, it's very possible that you would either be denied entry, or possibly given entry, but only for a period much shorter than 90 days.

If you were entering on a visa (which generally allows a ~6 months stay) they would potentially be a little more lenient in the situation described above, but it's still very possible they would give a shorter period of stay for the second trip - otherwise you could be spending up to 9 out of a 10 month period in the US which is very much the definition of "living" in the US.

This isn't a hard-and-fast rule, and will depend on the exact circumstances. The fact you are first visiting for leisure and then subsequently for business MAY mean they are a little less concerned. The fact you have a partner in the US MAY mean they are MORE concerned that you are planning to live in the US. It will all depend on the exact circumstances, the moods the border staff are in the at the time, and even things like your body language and how truthful you are with them (eg, if you try and hide the fact you have a partner in the US and they find out, the odds of you being denied entry increase significantly)

So putting that all together, a personal trip won't necessarily "use up" your 6 months (actually 180 days) as such, but depending on the length of the personal trip and the time between that trip and the business trip, it certainly could result in you being granted a duration of status on your second trip of less than 180 days.

There is also a 2nd issue you need to consider, which is the IRS's "Substantial Presence Test" which controls whether you are considered a United States resident for tax purposes. If you spend more than 183 days in the US in any calendar year, you are deemed a resident for tax purposes and will need to (at least) lodge a tax return. If you spend less than 183 days, but spend time in the preceding 2 years you may also fall into this category.

Yes, a traveler with a B-1 visa who is eligible for the VWP can use the VWP for visits that do not have a business purpose. However, your visa is a B-1/B-2 visa, so you shouldn't use the VWP at all. Your visa permits both personal visits and business visits. When you enter the US, you have to tell the officer the purpose of your trip, and the officer would then admit you either in B-1 or B-2 status, as appropriate.

In theory, the VWP is only for people who don't have an appropriate visa, because if they do have one, there is no need to waive the visa requirement. Sometimes, though, people who have a visa are admitted erroneously on the VWP.

In any event, you would realize no benefit by visiting under the VWP. Suppose you spend ten weeks in the US visiting your partner, and then you leave for a month, and then you return and seek entry in B-1 status for some business meetings. The decision to admit you will be the same regardless of whether previous visit was with a B visa or the VWP. On the other hand, there are some potential disadvantages to using the VWP, most notably that you cannot extend your stay if your circumstances change.

The other answer shows some misunderstanding of the time limits associated with the B visa. When you are admitted on a B visa, you will be admitted for at least six months (not 180 days) unless there is some unusual circumstance. You can actually be admitted for up to one year. Furthermore, you can apply to extend your stay in six-month increments. As the other answer correctly notes, this can expose you to income tax liability, and you also have to be able to explain how you can manage to spend so much time in the United States without doing any work, but it is theoretically possible.

Furthermore, if you visit for personal reasons and then suddenly have a need to attend a business meeting, you can apply to change your status from B-2 to B-1. In practice, that will probably take too long to be useful, and you may be better off leaving the country and returning. The application to change or extend nonimmigrant status is also somewhat expensive, carrying as it does a fee of $370.

Under the VWP, by contrast, you cannot extend your stay, nor can you change to another status.

The six-month minimum period of initial admission is specified at 8 CFR 214.2:

(b) Visitors -

(1) General. Any B-1 visitor for business or B-2 visitor for pleasure may be admitted for not more than one year and may be granted extensions of temporary stay in increments of not more than six months each...

(2) Minimum six month admissions. Any B-2 visitor who is found otherwise admissible and is issued a Form I-94 (see § 1.4), will be admitted for a minimum period of six months, regardless of whether less time is requested, provided, that any required passport is valid as specified in section 212(a)(26) of the Act. Exceptions to the minimum six month admission may be made only in individual cases upon the specific approval of the district director for good cause.

Shorter periods of admission are generally reserved for cases in which someone has been in the US recently, or in which the visitor's passport expires before the end of the normal period of admission (or in less than six months after the end of that period, depending on the country issuing the passport). The situation for the VWP is not quite so obvious from looking at the statute and the regulations, but you can see that it is essentially similar by looking at the bottom of page 116 of the CBP Inspector's Field Manual:

All VWP admissions are for 90 days unless the applicant’s passport is valid for a lesser period...

(The case of someone returning to the US before the expiration of their prior period of admission is covered elsewhere.)

So, to summarize for your case:

Since you have a B-1/B-2 visa, when you visit your partner and plan no business activities during the visit, you should request admission in B-2 status. But entering with the visa to visit your partner will not have any more negative effect on your future business visits than would entering on the VWP.